BY:ABHIJEET

SINGHX ‘A’

CONTENT1. Triangle and its classification

on the basis of length of sides on the basis of angle

2. Some properties of triangles

3. Congruent triangles and criteria for congruency of triangles

4. Similar Figures

5. Similar Triangles

6. BPT / Thales Theorem

7. Criteria for similarity of triangles

8. Areas of similar triangles

9. Pythagoras theorem

10. Summary

TriangleA closed figure consisting of three line segments linked end-to-end is called a triangle.

Scalene Triangle

•Scalene triangles are triangles in which none of the sides are the same length.

Isosceles

Triangle

•Isosceles triangles are triangles in which two of the sides are the same length.

Equilateral Triangle

•Equilateral triangles are triangles in which all three sides are the same length.

Types of triangles

on the basis of

length of sides

Acute Triangle

• Acute triangles are triangles in which the measures of all three angles are less than 90 degrees.

Obtuse

Triangle

• Obtuse triangles are triangles in which the measure of one angle is greater than 90 degrees.

Right Triangle

• Right triangles are triangles in which the measure of one angle equals 90 degrees.

Types of triangles on the basis of measure of

angles

1. All the medians of a triangle are concurrent. The point at which all the medians meet is known as centroid. Centroid divides the medians in the ratio 2:1.

2. All the altitudes of a triangle are concurrent. The point at which all the altitudes meet is known as orthocenter.

3. All the bisectors of angles of a triangle are concurrent. The point at which all the bisectors of triangle meet is known as incenter.

4. All the perpendicular bisectors of sides of a triangle are concurrent. The point at which all the perpendicular bisector of sides meet is known as circumcenter.

Triangle Inequality Theorem

The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

Area of triangle

• Area = ½ × base × height

• Area = √s (s-a)(s-b) (s-c)

• Area of equilateral triangle =

Heron’s Formula

{a,b and c are lengths of sides}

Congruent Triangles

Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

Criteria for congruency

 1. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

2. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

4. RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and pair of shorter sides is equal in length, then the triangles are congruent.

Similar FiguresThe figures which have the same shape but not necessarily of the same size, are called similar figures. Figures are said to be similar if :• They are enlargements of each other• Their corresponding angles are equal• Their corresponding sides are proportional

Example :(i) Every circle is similar to another circle.

(ii) Every square similar to another square

(iii) A right angle triangle is similar to any other right angle triangle

All congruent figures are similar but the similar figures neednot be congruent.

Similar TrianglesTwo triangles are similar, if(1) their corresponding angles are equal(2) their corresponding sides are in the same ratio (or proportion).

Equiangular triangles: If corresponding angles of two triangles are equal, then they are known as equiangular triangles.

The ratio of any two corresponding sides in two equiangular triangles is always the same.

THALES

Born : c. 624 BC

Died : c. 546 BC

Era : Pre-Socratic philosophy

Region : Western Philosophy

School : Ionian/Milesian school, Naturalism

Basic Proportionality Theorem(Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect theother two sides in distinct points, the other two sides are divided in the same Ratio.

Criteria for similarity of triangles

AAA Similarity Criterion: If in two triangles, the corresponding angles are equal, then their corresponding sides are proportional and hence the triangles are similar.

Corollary: If two angles of a triangle are respectively equal to two angles of another triangle, then the two triangles are similar by AA Similarity Criterion.

SSS Similarity Criterion: If the corresponding sides of two triangles are proportional their corresponding angles are equal and hence the triangles are similar.

In the given triangles we observe that the ratio of the corresponding sides is the same.

3/6 = 4/8 = 5/10 = 1/2

SAS Similarity Criterion: If one angle of a triangle is equal to one angle of the other and the sides including these angles are proportional, the triangles are similar.

Property: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

Answer the following questions dealing with similar figures.

1. Which of the following triangles are always similar?(a)  right triangles(b)  isosceles triangles(c)  equilateral triangles

 2. The sides of a triangle are 5, 6 and 10.  Find the length of the longest side of a similar triangle whose shortest side is 15.(a) 10 (b) 15(c) 18 (d) 30 3. Similar triangles are exactly the same shape and size. (a) True (b) False 

4. Given:  In the diagram, is parallel to ,  BD = 4, DA = 6  and EC = 8.  Find BC to the nearest tenth. (a) 4.3 (b) 5.3 (c) 8.3 (d) 13.3

5. Find BC. (a) 4  (b) 4.5  (c) 13.5  (d) 17

6. Two triangles are similar.  The sides of the first triangle are 7, 9, and 11.  The smallest side of the second triangle is 21.  Find the perimeter of the second triangle. (a) 27(b) 33(c) 63(d) 81

7. Two ladders are leaned against a wall such that they make the same angle with the ground.  The 10' ladder reaches 8' up the wall.  How much further up the wall does the 18' ladder reach?(a) 4.5'  (b) 6.4'  (c) 14.4'  (d) 22.4‘ 

8. At a certain time of the day, the shadow of a 5' boy is 8' long.  The shadow of a tree at this same time is 28' long.  How tall is the tree?(a) 8.5'  (b) 16'  (c) 17.5'  (d) 20‘

9. Two triangular roofs are similar.  The corresponding sides of these roofs are 16 feet and 24 feet.  If the altitude of the smaller roof is 6 feet, find the corresponding altitude of the larger roof. (a) 6(b) 9(c) 36(d) 81

10. Two polygons are similar.  If the ratio of the perimeters is 7:4, find the ratio of the corresponding sides. (a) 7 : 4 (b) 49 : 16

11. The ratio of the perimeters of two similar triangles is 3:7. Find the ratio of the areas. (a) 3 : 7 (b) 9 : 49

12. The areas of two similar polygons are in the ratio 25:81.  Find the ratio of the corresponding sides. (a) 5 : 9(b) 25 : 81(c) 625 : 6561

13. Given:  These two regular polygons are similar. What is their ratio of similitude (larger to smaller)?(a) 8 : 12(b) 2 : 1(c) 4 : 8

14. Given:  These two rhombi are similar. If their ratio of similitude is 3 : 1, find x.(a) 4(b) 6(c) 9

16. Given angle A and angle A' are each 59º, find AC.(a) 8(b) 10(c) 12(d) 18 17. The side of one cube measures 8 inches.  The side of a smaller cube measures 6 inches.  What is the ratio of the volumes of the two cubes (larger to smaller)?(a) 8:6(b) 16:9(c) 64:27(d) 64:36

18. In triangle ABC, angle A = 90º and angle B = 35º.  In triangle DEF, angle E = 35º and angle F = 55º.  Are the triangles similar? (a) YES(b) NO

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Given: Δ ABC ~ Δ PQR

To prove:

Construction: Draw AD ⊥ BC and PM ⊥ PR

Proof:

Areas of similarity triangles

Hence;

Now, in Δ ABD and Δ PQM; ∠ A = ∠ P, ∠ B = ∠ Q and ∠ D = ∠ M (because Δ ABC ~ Δ PQR)Hence; Δ ABD ~ Δ PQM Hence;

Since, Δ ABC ~ Δ PQR So,

Hence;

THEOREM:If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Construction: Triangle ABC is drawn which is right-angled at B. From vertex B, perpendicular BD is drawn on hypotenuse AC.

To prove: Δ ABC ~ Δ ADB ~ Δ BDC

Proof: In Δ ABC and Δ ADB; ∠ ABC = ∠ ADB∠ BAC = ∠ DAB ∠ ACB = ∠ DBA From AAA criterion; Δ ABC ~ Δ ADBIn Δ ABC and Δ BDC;

∠ ABC = ∠ BDC∠ BAC = ∠ DBC ∠ ACB = ∠ DBCFrom AAA criterion; Δ ABC ~ Δ BDC Hence; Δ ABC ~ Δ ADB ~ Δ BDC proved.

Pythagoras

Born: c. 570 BC Samos

Died: c. 495 BC (aged around 75) Metapontum

Era: Ancient Philosophy

Region: Western philosophy

School: Pythagoreanism

Pythagoras Theorem

Construction: Triangle ABC is drawn which is right angled at B. From vertex B, perpendicular BD is drawn on hypotenuse AC.

To prove: Proof:

In Δ ABC and Δ ADB;

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Because these are similar triangles

….(1)

In Δ ABC and Δ BDC;

Adding equations (1) and (2), we get;

Proved.

….(2)

1. Two figures having the same shape but not necessarily the same size are called similar figures.

2. All the congruent figures are similar but the converse is not true.

3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion).

4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).

Summary

7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion).

8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion).

9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion).

10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.

12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem).

13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.