1 eso - unit 10 - lines and angles. geometric figures
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Unit 10 April
1. LINES.
1.1. PERPENDICULAR BISECTOR.
In Geometry, Bisection is the division of something into two equal parts, usually
by a Line, which is then called a Perpendicular Bisector or Segment Bisector (a line
that passes through the Midpoint of a given segment).
How to construct a Perpendicular Bisector in a Segment π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½?
β’ Step 1: Stretch your compasses until it is more than half the length of π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½. Put the
sharp end at π¨π¨ and mark an arc above and another arc below line segment π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½.
β’ Step 2: Without changing the width of the compasses, put the sharp end at π¨π¨ and
mark arcs above and below the line segment π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½ that will intersect with the arcs
drawn in step 1.
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β’ Step 3: Join the two points where the arcs intersect with a straight line. This line is
the perpendicular bisector of π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½. P is the midpoint of π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½.
π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½ = π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½
π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½ = π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½οΏ½
Q
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1.2. ANGLE BISECTOR.
The Angle Bisector is a line that passes through the apex of an angle, that
divides it into two equal angles.
How to construct an Angle Bisector for the following angle?
β’ Step 1: Put the sharp end of your compasses at point π¨π¨ and make one arc on the
line π¨π¨π©π©οΏ½οΏ½οΏ½οΏ½ (point πΊπΊ) and another arc on line π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½ (point π»π»).
β’ Step 2: Put the sharp end of the compasses at πΊπΊ and make an arc within the
lines π¨π¨π¨π¨οΏ½οΏ½οΏ½οΏ½ and π¨π¨π©π©οΏ½οΏ½οΏ½οΏ½. Do the same at T and make sure that the second arc intersects
the first arc.
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β’ Step 3: Draw a line from point π¨π¨ to the points of intersection of the 2 arcs. This
line bisects π¨π¨π¨π¨π©π©οΏ½ .
MATH VOCABULARY: Bisection, Perpendicular Bisector, Segment Bisector, Arc,
Midpoint, Segment, Line, Apex.
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2. ANGLES.
2.1. PARALLEL ANGLES.
Two Parallel Angles are equal or Supplementary Angles.
Two angles are Supplementary when their addition is ππππππΒ°.
When Parallel Lines get crossed by another line (which is called a Transversal),
you can see that many angles are the same.
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The pairs of angles on opposite sides of the transversal but outside the two
lines are called Alternate Exterior Angles.
The angles in matching corners are called Corresponding Angles.
The pairs of angles on opposite sides of the transversal but inside the two lines
are called Alternate Interior Angles.
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The pairs of angles on one side of the transversal but inside the two lines are
called Consecutive Interior Angles.
Vertically Opposite Angles are the angles opposite each other when two lines
cross.
Alternate Exterior Angles, Corresponding Angles, Alternate Interior Angles and
Vertically Opposite Angles are equal.
MATH VOCABULARY: Parallel Angles, Supplementary Angles, Transversal, Alternate
Exterior Angles, Corresponding Angles, Alternate Interior Angles, Consecutive Interior
Angles, Vertically Opposite Angles.
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2.2. MEASUREMENT OF ANGLES.
2.2.1. SEXAGESIMAL DEGREE.
Remember that a Right Angle is ππππΒ°.
A Straight Angle is ππππππ π π π π π π π π π π π π π π .
A Full Angle is ππππππ π π π π π π π π π π π π π π .
There are ππππππ π π π π π π π π π π π π π π in one Full Rotation (one complete circle around). The
most usual unit of measurement for angles is the Sexagesimal Degree, which consists
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in ππ/ππππππ of a full angle. The measurement of an angle in Sexagesimal Degrees is
denoted by the symbol Β°. To get an idea, one degree corresponds to the following
aperture:
What happens when we have an angle of less than ππΒ°?
To be able to speak about angles that measure less than ππΒ°, we use
submultiples of a degree, so we avoid working with expressions like the following:
β’ This angle measures half a degree.
β’ This angle measures 0.56 degrees
Thus, the Sexagesimal Degree has submultiples: these are the Minute and the
Second. The Minute is designated as β and the second as ββ.
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The measurement of an angle in degrees, minutes and seconds would be, for
example, ππππΒ° ππππβ² ππππβ²β². It would be read as: an angle of ππππ degrees, ππππ minutes and
ππππ seconds.
Let's see the exact value of minutes and seconds:
β’ One Minute is the result of taking a degree and dividing ππππ it into equal parts. This
is, mathematically expressed:
πππππππππππππ π =ππΒ°ππππ
β ππππ πππππππππππ π π π = ππΒ°
β’ A Second is the result of taking a minute and dividing it in ππππ equal parts. This is,
mathematically expressed:
ππ π π π π πππππππ π =ππβ²ππππ
β ππππ π π π π πππππππ π π π = ππβ²
2.2.2. USING A PROTRACTOR.
To find the number of degrees in an angle, we use a Protractor.
A straight angle is divided into ππππππ equal subdivisions on a Protractor, marked
from ππ to ππππππ. You find two scales marked ππ to ππππππ, one in clockwise direction and
the other in anti-clockwise direction. Each subdivision stands for ππΒ°.
Steps to measure:
β’ Place the center of the protractor on the vertex of the angle.
β’ Base line should fall along any of the sides.
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β’ The scale (clock wise or anti-clock wise direction), which begins with zero on the
side, is chosen.
β’ Read the mark on the scale where the other arm crosses it.
2.2.3. COMPLEX AND NON-COMPLEX FORM.
To change degrees into minutes and seconds we will always work by means of
Conversion Factors. To convert from Complex Form to Non-Complex Form:
π΄π΄πππππππππ π π π = π«π«π π π π π π π π π π π π β ππππ
πΊπΊπ π πππππππ π π π = π΄π΄πππππππππ π π π β ππππ
πΊπΊπ π πππππππ π π π = π«π«π π π π π π π π π π π π β ππ,ππππππ
Write 73Β° 13β² 48β²β² in seconds:
73Β° = 73 β 3,600β²β² = 262,800β²β²
13β² = 13 β 60β²β² = 780´´
73Β° 13β² 48β²β² = 262,800β²β² + 780β²β² = 263,628β²β²
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To do it backwards:
π΄π΄πππππππππ π π π =πΊπΊπ π πππππππ π π π
ππππ;πΉπΉπ π πππΉπΉπππππ π π π π π = β²β²
π«π«π π π π π π π π π π π π =π΄π΄πππππππππ π π π
ππππ;πΉπΉπ π πππΉπΉπππππ π π π π π = β²
π«π«π π π π π π π π π π π π =πΊπΊπ π πππππππ π π π ππ,ππππππ
;πΉπΉπ π πππΉπΉπππππ π π π π π = β²β²
Write 263,628β²β² in complex form:
263,628β²β² Γ· 3,600 = 73Β°,π π π π π π π π π π π π π π π π π π = 828´´
828´´ Γ· 60 = 13β²,π π π π π π π π π π π π π π π π π π = 48β²β²
Therefore:
263,628β²β² = 73Β° 13β² 48β²β²
MATH VOCABULARY: Right Angle, Straight Angle, Sexagesimal Degree, Full Angle,
Minute, Second, Protractor.
2.3. ANGLES OPERATIONS.
2.3.1. ADDING ANGLES USING THE SEXAGESIMAL MEASURE.
To Add we need to add separately degrees or hours, minutes and seconds and
then convert the seconds into minutes and the minutes into degrees/hours if we get
more than ππππ.
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Add 15Β° 43β² 30β²β² + 25Β° 50β² 34β²β²
15Β° 43β² 30β²β²+ 25Β° 50β² 34β²β²
40Β° 93β² 64β²β²+1β² β60β²β²
40Β° 94β² 4β²β²+1Β° β60β²41Β° 34β² 4β²β²
2.3.2. SUBTRACTING ANGLES USING THE SEXAGESIMAL MEASURE.
To Subtract we need to subtract separately degrees/hours, minutes and
seconds, if we do not have enough seconds or minutes we convert one degree/hour
into minutes or a minute into seconds.
Subtract 56Β° 38β²11β²β² β 32Β° 43β² 56β²β²
56Β° 38β² 11β²β²β 32Β° 43β² 56β²β²
56Β° 38β² 11β²β²β1Β° +ππππβ² +60β²β²
55Β° 97β² 71β²β²β 32Β° 43β² 56β²β²
23Β° 54β² 15β²β²
We obtain ππππβ² = +60β² β 1β², 60β² from subtracting 1Β° from degrees and 1β² by
given +60β²β² to seconds.
2.3.3. MULTIPLICATION BY A NATURAL NUMBER.
We Multiply separately degrees/hours, minutes and seconds and then we
convert the seconds into minutes and the minutes into degrees/hours when we get
more than ππππ subunits.
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Multiply (13Β° 23β² 26β²β²) Β· 4:
13Β° 23β² 26β²β²π₯π₯ 452Β° 92β² 104β²β²
+1β² β60β²β²52Β° 93β² 44β²β²+1Β° β60β²53Β° 33β² 44β²β²
2.3.4. DIVISION BY A NATURAL NUMBER.
We Divide the degrees/hours, and the remainder is converted into minutes
that must be added to the previous quantity that we had, divide the minutes and we
repeat the same that we have before. The remainder is in seconds.
2.4. INTERIOR ANGLES OF POLYGONS.
An Interior Angle is an angle inside a Shape.
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2.4.1. TRIANGLES.
The Interior Angles of a Triangle add up to ππππππΒ°
60Β° + 90Β° + 30Β° = 180Β°
70Β° + 80Β° + 30Β° = 180Β°
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2.4.2. QUADRILATERALS.
A Quadrilateral has 4 straight sides. The Interior Angles of a Quadrilateral add
up to ππππππΒ°.
90Β° + 90Β° + 90Β° + 90Β° = 360Β°
100Β° + 80Β° + 90Β° + 90Β° = 360Β°
Because there are 2 triangles in a square: ππππππΒ° + ππππππΒ° = ππππππΒ°
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2.4.3. PENTAGON.
A Pentagon has 5 sides, and can be made from three triangles. Its interior
angles add up to ππ Γ ππππππΒ° = ππππππΒ°. And when it is regular (all angles the same),
then each angle is ππππππΒ° / ππ = ππππππΒ°.
The Interior Angles of a Pentagon add up to ππππππΒ°.
2.4.4. THE GENERAL RULE.
Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon,
etc), we add another ππππππΒ° to the total:
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πΊπΊππππ ππππ π°π°πππππ π π π πππππ π π¨π¨πππ π π¨π¨π π π π = (ππ β ππ) Γ ππππππΒ°
π¬π¬πΉπΉπππ¬π¬ π¨π¨πππ π π¨π¨π π (ππππ πΉπΉ πΉπΉπ π π π πππ¨π¨πΉπΉπ π π¨π¨πππ¨π¨π·π·π π ππππ) = (ππ β ππ) Γ ππππππΒ°
ππ
2.5. CIRCUMFERENCE ANGLES.
2.5.1. CENTRAL ANGLE.
Central Angle is the angle subtended at the center of a circle by two given
points on the circle.
2.5.2. INSCRIBED ANGLE.
Inscribed Angle is an angle made from points sitting on the circle's
circumference.
π¨π¨ and π©π© are "End Points"
π¨π¨ is the "Apex Point"
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2.5.3. INSCRIBED ANGLE THEOREMS.
An Inscribed Angle πΉπΉΒ° is half of the Central Angle πππΉπΉ. Called The Angle at the
Center Theorem.
And (keeping the endpoints fixed) the angle πΉπΉΒ° is always the same, no matter
where it is on the circumference the angle πΉπΉΒ° is the same. Called The Angles
Subtended by Same Arc Theorem.
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2.5.4. ANGLE IN A SEMICIRCLE.
An Angle Inscribed in a Semicircle is always a Right Angle. The end points are
either end of a circle's diameter, the apex point can be anywhere on the
circumference.
The inscribed angle ππππΒ° is half of the central angle ππππππΒ°.
MATH VOCABULARY: Polygon, Interior Angle, Shape, Triangle, Quadrilateral, Square,
Side, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon, Circumference,
Central Angle, Circle, Arc, Inscribed Angle, End Points, Apex Point, Theorem, Semicircle.
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3. LINES OF SYMMETRY OF PLANE SHAPES.
3.1. LINE OF SYMMETRY.
Another name for Reflection Symmetry. One half is the reflection of the other
half. The "Line of Symmetry" is the imaginary line where you could fold the image and
have both halves match exactly.
You can find if a shape has a Line of Symmetry by folding it. When the folded
part sits perfectly on top (all edges matching), then the fold line is a Line of Symmetry.
Here I have folded a rectangle one way, and it didn't work.
But when I try it this way, it does work (the folded part sits perfectly on top, all
edges matching):
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3.2. TRIANGLES.
A Triangle can have ππ, or ππ or ππππ π¨π¨πππππ π π π of symmetry:
3.3. QUADRILATERALS.
Different types of Quadrilaterals (a 4-sided plane shape):
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3.4. REGULAR POLYGONS.
A Regular Polygon has all sides equal, and all angles equal:
And the pattern continues:
β’ A regular polygon of 9 sides has 9 Lines of Symmetry
β’ A regular polygon of 10 sides has 10 Lines of Symmetry
β’ ...
β’ A regular polygon of "n" sides has "n" Lines of Symmetry
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3.5. CIRCLE.
A line (drawn at any angle) that goes through its center is a Line of Symmetry.
So a Circle has infinite Lines of Symmetry.
3.6. PLANE SHAPES.
If a Plane Shape has ππ Lines of Symmetry, all of then cut in one point, and
every two near lines form an angle:
πΆπΆ =ππππππΒ°ππ
πΆπΆ =ππππππΒ°ππ
= ππππΒ°
MATH VOCABULARY: Symmetry, Plane Shapes, Line of Symmetry, Reflection
Symmetry, Rhombus.
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4. PLANE FIGURES.
4.1. TRIANGLES.
4.1.1. EQUILATERAL, ISOSCELES AND SCALENE.
There are three special names given to triangles that tell how many sides (or
angles) are equal. There can be 3, 2 or no equal sides/angles.
4.1.2. ACUTE, OBTUSE AND RIGHT ANGLES.
Triangles can also have names that tell you what type of angle is inside:
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4.1.3. RELATIONSHIP OF SIDES TO INTERIOR ANGLES IN A TRIANGLE.
In a Triangle:
β’ The shortest side is always opposite the smallest interior angle.
β’ The longest side is always opposite the largest interior angle.
Recall that in a Scalene Triangle, all the sides have different lengths and all the
interior angles have different measures. In such a triangle, the shortest side is always
opposite the smallest angle. (These are shown in bold color above)
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Similarly, the longest side is opposite the largest angle. If the smallest side is
opposite the smallest angle, and the longest is opposite the largest angle, then it
follows that since a triangle only has three sides, the midsize side is opposite the
midsize angle.
An Equilateral Triangle has all sides equal in length and all interior angles equal.
Therefore there is no "largest" or "smallest" in this case.
Isosceles Triangles have two sides the same length and two equal interior
angles. Therefore there can be two sides and angles that can be the "largest" or the
"smallest".
4.1.4. CONSTRUCT A TRIANGLE GIVEN THE LENGTH OF ITS THREE SIDES.
We can use a pair of compasses and a ruler to construct a triangle when the
lengths of its sides are given.
Construct a triangle βππππππ with πππποΏ½οΏ½οΏ½οΏ½ = π±π± ππππ, πππποΏ½οΏ½οΏ½οΏ½ = π²π² ππππ and πππποΏ½οΏ½οΏ½οΏ½ = π³π³ ππππ.
β’ Step 1: Draw a line segment πππποΏ½οΏ½οΏ½οΏ½ with lengths π±π± ππππ and mark the points π¨π¨ and π¨π¨.
β’ Step 2: To draw πππποΏ½οΏ½οΏ½οΏ½, first stretch the 2 arms of the compasses π²π² ππππ apart, place
the sharp point at point π¨π¨ and mark an arc with the pencil end.
x cm
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β’ Step 3: To draw πππποΏ½οΏ½οΏ½οΏ½, adjust the compasses to π³π³ ππππ, place the sharp point at
point π¨π¨ and mark an arc with the pencil end. You need to draw the arc so that it
will intersect with the arc drawn in step 2. Label the point of intersection as
point πΉπΉ.
β’ Step 4: Draw straight lines from π¨π¨ to πΉπΉ and from π¨π¨ to πΉπΉ.
x cm
x cm
x cm
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4.1.5. MEDIAN. CENTROID.
A Median of a Triangle is a line segment from a vertex (corner point) to the
midpoint of the opposite side.
A triangle has three medians, and they all cross over at a special point called
the "Centroid".
4.1.6. ALTITUDE. ORTHOCENTER.
Altitude is another word for height. In a triangle is the line at right angles to a
side that goes through the opposite corner.
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Where all three lines intersect is the "Orthocenter":
Note that sometimes the edges of the triangle have to be extended outside the
triangle to draw the Altitudes. Then the Orthocenter is also outside the triangle.
4.1.7. CIRCUMCENTER.
Using the Perpendicular Bisector of sides:
Where all three perpendicular bisectors intersect is the center of a triangle's
"Circumcircle", called the "Circumcenter":
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4.1.8. INCENTER.
If we use the Angle Bisector of the angles of the triangle:
Where all three angle bisectors intersect is the center of a triangle's "Incircle",
called the "Incenter"
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4.1.9. PYTHAGORAS' THEOREM.
When a triangle has a Right Angle (90Β°), and squares are made on each of the
three sides, then the biggest square has the exact same area as the other two squares
put together!
It is called "Pythagoras' Theorem" and can be written in one short equation
Note: ππ is the longest side of the triangle, πΉπΉ and ππ are the other two sides.
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The longest side of the triangle is called the "Hypotenuse", so the formal
definition is:
In a right angled triangle: the square of the hypotenuse is equal to the sum of the
squares of the other two sides.
52 = 32 + 42
25 = 9 + 16 = 25
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There are many geometric problems where we have to use the Pythagoras'
Theorem:
We have to study the shape to find out the right angles and the right triangles,
and apply the Pythagoras' Theorem:
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Rhombi
Trapezoids
Regular Polygons
Circles
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4.1.10. AREA AND PERIMETER.
The Perimeter is the distance around the edge of the triangle: just add up the
three sides:
208 + 145 + 203 = 556
The Area is half of the base times height. "ππ" is the distance along the Base. "π¬π¬"
is the height (measured at right angles to the base).
π¨π¨ =ππ β π¬π¬ππ
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Many times we will have to use the Pythagoras' Theorem to find the height of
the triangle.
β2 + 52 = 102
β2 = 102 β 52 = 100 β 25 = 75
β = β75 β 8.7 πππ π
π΄π΄ =10 β 8,7
2= 43.5 πππ π 2
MATH VOCABULARY: Plane Figures, Equilateral, Isosceles, Scalene, Acute, Obtuse,
Median, Centroid, Altitude, Orthocenter, Circumcenter, Incenter, Pythagoras' Theorem,
Hypotenuse, Rhombus, Trapezoid, Area, Perimeter, Base.
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4.2. QUADRILATERALS.
Quadrilateral just means "four sides" (quad means four, lateral means side). A
Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join
up), and has straight sides.
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4.2.1. SQUARE.
A Square is a flat shape with 4 equal sides and every angle is a right angle (90Β°).
Properties:
β’ All sides are equal in length.
β’ Each internal angle is 90Β°.
β’ Opposite sides are parallel (so it is a Parallelogram).
The Area is the side length squared:
π¨π¨π π π π πΉπΉ = πΉπΉ β πΉπΉ = πΉπΉππ
The Area is also half of the diagonal squared:
π¨π¨π π π π πΉπΉ =π π ππ
ππ
The Perimeter is 4 times the side length:
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π¨π¨ = πΉπΉ + πΉπΉ + πΉπΉ + πΉπΉ = ππ β πΉπΉ
A square has two Diagonals; they are equal in length and intersect in the
middle. The Diagonal is the side length times the square root of 2:
π π = πΉπΉ β βππ
4.2.2. RECTANGLE.
A Rectangle is a four-sided flat shape where every angle is a right angle (90Β°).
Properties:
β’ Each internal angle is 90Β°.
β’ Opposite sides are parallel (so it is a Parallelogram).
The Area is the width times the height:
π¨π¨ = ππ β π¬π¬
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The Perimeter is the distance around the edges. The Perimeter is 2 times the
(width + height):
π¨π¨ = ππππ + πππ¬π¬ = ππ(ππ + π¬π¬)
A rectangle has two Diagonals, they are equal in length and intersect in the
middle.
The Diagonal is the square root of (width squared + height squared):
π π = οΏ½ππππ + π¬π¬ππ
4.2.3. RHOMBUS.
A Rhombus is a flat shape with 4 equal straight sides.
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Properties:
β’ All sides have equal length.
β’ Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram).
β’ The Altitude is the distance at right angles to two sides.
β’ And the Diagonals "ππ" and "ππ" of a rhombus bisect each other at right angles.
β’ The Square is a Rhombus.
The Area can be calculated by:
β’ The altitude times the side length:
π¨π¨ = πΉπΉπ¨π¨πππππππππ π π π β π π
β’ Multiplying the lengths of the diagonals and then dividing by 2:
π¨π¨ = ππ β ππππ
The Perimeter is 4 times "s" (the side length) because all sides are equal in
length:
π¨π¨ = ππ β π π
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It is more common to call this shape a Rhombus, but some people call it a
Rhomb or even a Diamond. The plural is Rhombi or Rhombuses, and, rarely, Rhombbi
or Rhombbuses (with a double b).
4.2.4. PARALLELOGRAM.
A Parallelogram is a flat shape with opposite sides parallel and equal in length.
Properties:
β’ Opposite sides are parallel.
β’ Opposite sides are equal in length.
β’ Opposite angles are equal (angles "πΉπΉ" are the same, and angles "ππ" are the same).
β’ Angles "πΉπΉ" and "ππ" add up to ππππππΒ°, so they are supplementary angles.
β’ A Parallelogram where all angles are right angles is a rectangle.
The Area is the base times the height:
π¨π¨ = ππ β π¬π¬
The Perimeter is 2 times the (base + side length):
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π¨π¨ = ππππ + πππ π = ππ(ππ + π π )
The Diagonals of a parallelogram bisect each other. In other words the
diagonals intersect each other at the half-way point.
4.2.5. TRAPEZOID.
A Trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite
sides parallel (marked with arrows below):
Properties:
β’ Has a pair of parallel sides.
β’ Is an Isosceles Trapezoid when both angles coming from a parallel side are equal,
and the sides that aren't parallel are equal in length.
β’ Is called a "Trapezium" in the UK.
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The Area is the average of the two base lengths times the altitude:
π¨π¨ =πΉπΉ β ππππ
β π¬π¬
The Perimeter is the sum of all side lengths:
π¨π¨ = πΉπΉ + ππ + ππ + π π
The Median (also called a midline or Midsegment) is a line segment half-way
between the two bases.
ππ =πΉπΉ + ππππ
You can calculate the area when you know the median, it is just the median
times the height:
π¨π¨ = ππ β π¬π¬
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A Trapezium (UK: Trapezoid) is a quadrilateral with NO parallel sides. The US
and UK have their definitions swapped over.
4.2.6. KITE.
A Kite is a flat shape with straight sides. It has 2 pairs of equal adjacent sides.
Properties:
β’ Two pairs of sides.
β’ Each pair is made up of adjacent sides (they meet) that are equal in length.
β’ The angles are equal where the two pairs meet.
β’ Diagonals (dashed lines) cross at right angles, and one of the diagonals bisects
(cuts equally in half) the other.
β’ When all sides have equal length the Kite will also be a Rhombus.
β’ When all the angles are also ππππΒ° the Kite will be a Square.
To find the Area of a Kite, Multiply the lengths of the diagonals and then divide
by 2 to find the Area:
π¨π¨ =ππ β ππππ
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The Perimeter is 2 times (side length a + side length b):
π¨π¨ = ππ(πΉπΉ + ππ)
A concave Kite is called a Dart.
MATH VOCABULARY: Diagonal, Rectangle, Rhomb, Diamond, Rhombi, Rhombuses,
Rhombbi, Rhombbuses, Parallelogram, Trapezoid, Trapezium, Midsegment, Kite, Dart.
4.3. POLYGONS.
A Polygon is a plane shape with straight sides. Polygons are 2-dimensional
shapes. They are made of straight lines, and the shape is "closed" (all the lines connect
up).
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A Regular Polygon has all angles equal and all sides equal, otherwise it is
Irregular.
A Convex Polygon has no angles pointing inwards. More precisely, no internal
angle can be more than ππππππΒ°. If any internal angle is greater than ππππππΒ° then the
polygon is Concave. (Think: concave has a "cave" in it).
A Simple Polygon has only one boundary, and it doesn't cross over itself.
A complex polygon intersects itself! Many rules about polygons don't work when it is
Complex.
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4.3.1. CIRCUMCIRCLE, INCIRCLE, RADIUS AND APOTHEM OF REGULAR POLYGONS.
They are just the names of the "Outer" and "Inner" circles (and each Radius)
that can be drawn on a polygon like this:
The "outside" circle is called a Circumcircle, and it connects all vertices (corner
points) of the polygon. The Radius of the circumcircle is also the radius of the polygon.
The "inside" circle is called an Incircle and it just touches each side of the polygon at its
midpoint. The radius of the incircle is the Apothem of the polygon.
4.3.2. AREA AND PERIMETER OF REGULAR POLYGONS.
The Perimeter is ππ times the Side, where ππ is the number of sides.
π¨π¨ = π π β ππ
We can learn a lot about regular polygons by breaking them into triangles like
this:
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Notice that:
β’ The "Base" of the triangle is one Side of the polygon.
β’ The "Height" of the triangle is the "Apothem" of the polygon.
Now, the Area of a Triangle is half of the base times height, so:
π¨π¨ =ππ β π¬π¬ππ
=π π β πΉπΉπππ¬π¬πππππ π ππ
ππ
To get the area of the whole polygon, just add up the areas of all the little
triangles ("ππ" of them)
π¨π¨π π π π πΉπΉ ππππ π¨π¨πππ¨π¨π·π·π π ππππ =π π β πΉπΉπππ¬π¬πππππ π ππ
ππβ ππ =
π¨π¨ β πΉπΉπππ¬π¬πππππ π ππππ
MATH VOCABULARY: Convex, Concave, Apothem, Radius.
4.4. CIRCLE.
A Circle is easy to make: Draw a curve that is "Radius" away from a central
point. All points are the same distance from the center.
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4.4.1. AREA AND CIRCUMFERENCE.
The Radius is the distance from the center outwards. The Diameter goes
straight across the circle, through the center. The Circumference is the distance once
around the circle.
π©π©πππ π πππππππππ π π π π π πππππ π = π π β π«π«πππΉπΉπππ π πππ π π π
Also note that the Diameter is twice the Radius:
π«π«πππΉπΉπππ π πππ π π π = ππ β πΉπΉπΉπΉπ π πππππ π
π©π©πππ π πππππππππ π π π π π πππππ π = ππ β π π β πΉπΉπΉπΉπ π πππππ π
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The Area of a circle is π π times the radius squared, which is written:
π¨π¨ = π π β π π ππ
4.4.2. LINES AND SLICES.
A line that goes from one point to another on the circle's circumference is
called a Chord. If that line passes through the center it is called a Diameter. A line that
"just touches" the circle as it passes by is called a Tangent. And a part of the
circumference is called an Arc.
There are two main "Slices" of a circle. The "pizza" slice is called a Sector. And
the slice made by a chord is called a Segment
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The Quadrant and Semicircle are two special types of Sector: Quarter of a circle
is called a Quadrant. Half a circle is called a Semicircle.
You can work out the Area of a Sector by comparing its angle to the angle of a
full circle.
π¨π¨π π π π πΉπΉ ππππ πΊπΊπ π πππππππ π = ππ β ππ β π π ππ
ππππππ (πππ¬π¬π π ππ π½π½ πππ π ππππ π π π π π π π π π π π π π π )
The Area of a Segment is the area of a sector minus the triangular piece (shown
in light blue here).
π¨π¨π π π π πΉπΉ ππππ πΊπΊπ π π π πππ π ππππ = π¨π¨π π π π πΉπΉ ππππ πΊπΊπ π πππππππ π β π¨π¨π π π π πΉπΉ ππππ π»π»π π πππΉπΉπππ π π¨π¨π π
The Arc Length (of a Sector or Segment) is:
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π³π³ = ππ β π π β π π β π½π½
ππππππ (πππ¬π¬π π ππ π½π½ πππ π ππππ π π π π π π π π π π π π π π )
4.4.3. RELATIVE POSITIONS.
The Circle has different Relative Positions:
β’ Relative Position of a Straight Line with respect to a circumference:
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β’ Relative Position of two circumferences:
MATH VOCABULARY: Convex, Center, Diameter, Slice, Chord, Tangent, Arc, Sector,
Segment, Quadrant, Semicircle.
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5. SOLID GEOMETRY.
Solid Geometry is the geometry of three-dimensional space, the kind of space
we live in. It is called three-dimensional, or 3D, because there are three dimensions:
Width, Depth and Height.
Let us start with some of the simplest shapes:
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There are two main types of solids, "Polyhedra", and "Non-Polyhedra": The
Polyhedra must have flat faces. If not they are Non-Polyhedra.
5.1. POLYHEDRONS.
A Polyhedron is a solid with flat faces. So no curved surfaces: cones, spheres
and cylinders are not polyhedrons.
5.1.1. THE PLATONIC SOLIDS.
A Platonic Solid is a 3D shape where:
β’ Each face is the same Regular Polygon.
β’ The same number of polygons meet at each vertex (corner).
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5.1.2. PRISMS.
A Prism is a solid object with:
β’ Identical ends.
β’ Flat faces.
β’ The same cross section all along its length.
A Cross Section is the shape made by cutting straight across an object.
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The Cross Section of this object is a triangle it has the same cross section all
along its length so it's a Triangular Prism. The ends of a prism are parallel each one is
called a Base.
The side faces of a prism are Parallelograms (4-sided shapes with opposite
sides parallel).
These are all Prisms:
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All the previous examples are Regular Prisms, because the cross section is
regular (in other words it is a shape with equal edge lengths, and equal angles.). Here is
an example of an Irregular Prism:
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5.1.3. PYRAMIDS.
A Pyramid is made by connecting a base to an Apex:
There are many types of Pyramids, and they are named after the shape of their
Base.
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This tells us where the top (apex) of the pyramid is. When the apex is directly
above the center of the base it is a Right Pyramid, otherwise it is an Oblique Pyramid.
This tells us about the shape of the base. When the base is a regular polygon it
is a Regular Pyramid, otherwise it is an Irregular Pyramid.
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5.2. NON-POLYHEDRA.
5.2.1. SPHERE.
Properties:
β’ It is perfectly symmetrical.
β’ All points on the surface are the same distance "r" from the center.
β’ It has no edges or vertices (corners).
β’ It has one surface (not a "face" as it isn't flat).
β’ It is not a polyhedron.
β’ A Sphere is a Rotated Circle.
5.2.2. CYLINDER.
Properties:
β’ It has a flat Base and a flat top.
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β’ The base is the same as the top.
β’ From base to top the shape stays the same.
β’ It has one curved side.
β’ It is not a polyhedron as it has a curved surface.
β’ A Cylinder is a Rotated Rectangle.
5.2.3. CONE.
Properties:
β’ It has a flat base.
β’ It has one curved side.
β’ It is not a polyhedron as it has a curved surface.
β’ The pointy end of a cone is called the Apex.
β’ The flat part is the Base
β’ A Cone is a Rotated Triangle.
MATH VOCABULARY: 3D Shapes, Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Pyramid,
Prism, Polyhedra, Non-Polyhedra, Polyhedron, Platonic Solids, Edge, Face, Vertices,
Vertex, Apex.
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