Test Answers and Exam Booklet

Geometric Construction

ANSWERS TO THE GEOMETRIC CONSTRUCTION TEST . . . . . . . . . . . 1

SAMPLE ANSWERS TO THE GEOMETRIC CONSTRUCTION PROBLEMS. . . 3

ANSWERS TO THE MATH PROBLEMS . . . . . . . . . . . . . . . . . . . . 32

EXAMINATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Contents

iii

ANSWERS TO THE GEOMETRIC CONSTRUCTION TEST

1. What is known to be true about the opposite angles created by intersectinglines? The opposite angles created by intersecting lines are equal.

2. Define parallel lines. Parallel lines are lines that are the samedistance apart along their length. If they were to be extended toinfinity, parallel lines would never intersect.

3. Define perpendicular. A line that’s perpendicular to another linecreates an angle of 90°, also known as a right angle.

4. How many degrees are there in a circle? There are 360° in a circle.

5. How are angles sized? Angles are sized in degrees. A degree can bedivided into 60 minutes, and a minute into 60 seconds.

6. Name four basic types of angles. The four basic types of angles are thestraight angle, the right angle, the acute angle, and the obtuse angle.

7. Define triangle. A triangle is formed by three lines that intersect toform three separate angles. The sum of the interior angles of thisgeometric figure is always 180°.

8. Show how an angle is named [labeled]. There are several ways to des-ignate a particular angle. All of these ways rely on labeling differentparts of the figure. Frequently, the ends of the lines and the angle’svertex are labeled. Sometimes, though, an angle is identified by asingle letter or number that represents the opening of the anglerather than the angle’s vertex. Such is the case with angle 1 shownhere. By referring to the other labels, you can also identify thisshape as angle C, angle ACB, or angle BCA.

9. Name and show an example of three types of triangles. The three majortypes of triangles are right, acute, and obtuse. In the right triangle, oneinterior angle equals 90°. In the acute triangle, each interior angle isless than 90°. In the obtuse triangle, one interior angle is greaterthan 90°. Examples of these three types of triangles are shown here.

Geometric Construction

1

10. Show two methods of naming [labeling] triangles. Particular triangles aretypically designated either by labeling each vertex or by labelingeach side. In the examples shown here, the vertices of triangle XYZare labeled X, Y, and Z, and the sides of triangle xyz are labeled x,y, and z.

11. Name four particular examples of quadrilaterals. All of the followingare more descriptive names for particular quadrilaterals, or four-sided figures: parallelograms such as the square, the rectangle, therhombus, and the rhomboid; trapezoids; and trapeziums. Namingany four of these shapes correctly answers the question.

12. What name is given to the distance from the center to the circumference ofa circle? The radius is the distance from the center to the circumfer-ence of a circle.

13. What is the name of the distance that goes across a circle through its center?The diameter is the distance that goes across a circle through itscenter.

14. What is a circle called that touches the flats of a polygon? A circle thattouches the flats of a polygon is called an inscribed circle.

15. What is a circle called that touches the corners of a polygon? A circle thattouches the corners of a polygon is called a circumscribed circle.

16. Show and label an example of concentric and eccentric circles. Two circleswith different radii but that share the same center are called con-centric circles. Two circles drawn one within the other but withdifferent centers are called eccentric circles. Examples of concentricand eccentric circles are shown here.

2 Geometric Construction

17. Show an example of a line tangent to a circle and two circles tangent toeach other. When a line is tangent to a circle, that line touches thecircle at only one point; it doesn’t intersect the circle. When two circlesare tangent to each other, they touch at only one point. Examples ofthese two common types of tangencies are shown here.

18. When should templates be used to make circles and arcs? Such templatesshould be used whenever possible. They’re especially useful whendrafting arc tangencies.

19. When should templates be used to make elliptical shapes? To save timein drafting, templates should be used even if it means approximatingthe ellipse. How close to the intended shape you need to be dependson company standards.

20. Which of the following methods could be used to most easily produce circlesand arcs: compass, templates, computer graphics system? Such shapes couldbe most easily produced using a computer graphics system. Theprogramming in the system makes such tasks practically automatic.

SAMPLE ANSWERS TO THE GEOMETRIC CONSTRUCTIONPROBLEMS

Geometric Construction 3

32 Geometric Construction

ANSWERS TO THE MATH PROBLEMS

1. Find the interior angles of an octagon. The interior angles of anoctagon are each 135�.

2. Find the interior angles of an equilateral triangle. The interior anglesof an equilateral triangle are each 60�.

3. Find the interior angles of a pentagon. The interior angles of apentagon are each 108�.

4. Find the interior angles of a decagon. The interior angles of a deca-gon are each 144�.

5. Find the interior angles of a square. The interior angles of a squareare each 90�.

6. How many sides does a regular polygon have if its interior angle is 140�?In the question, the size of each of the shape’s interiorangles is identified as 140�. The following formula can be adaptedto calculate the number of sides (n) given the angle size (�).

As shown here, the question can be answered by substituting 140�

for � in the formula and solving for n.

n

n�

2180 �

Step 1. Because the formula is being used to solvefor n, you should write the formula sothat n is on the left of the equal sign.

n

n� �

2180 140 Step 2. Substitute 140� for �.

n

n�

�

�

�

2180

180

140

180

Step 3. Divide both sides of the equal signby 180�, and simplify the resulting expressionsby canceling.

nn

nn

2 7

9

Step 4. Begin solving for the n variable by cancel-ing out the variable in the denominator on theleft side of the equal sign. Do so by multiplyingthe expressions on both sides of the equal signby n.

9

92

7

9

7

9

7

9

n n n n Step 5. Express the equation so that the variablewill be only the left side of the equal sign by sub-

tracting7

9

nfrom both sides. (Let n =

9

9

non the left

side of the = sign.)

7

9

� �n

n

2180

Geometric Construction 33

2

92 0

n Step 6. Subtract.9

9

7

9

2

9and

7

9

7

9

n n n n n0

2

92

n Step 7. Furthur isolate the variable by adding 2on both sides of the equal sign.

9

2

2

92

9

2

n Step 8. Finish isolating the variable by multiply-ing the expresions on both sides of the equal sign

by9

2. Simplify by canceling. Answer: The regular

polygon has 9 sides.

NOTES

34 Geometric Construction

Geometric Construction

When you feel confident that you have mastered the material in this study unit, complete thefollowing examination. Then submit only your answers to the school for grading, using one of theexamination answer options described in your “Test Materials” envelope. Send your answers forthis examination as soon as you complete it. Do not wait until another examination is ready.

Questions 1–20: Select the one best answer to each question.

1. The line that starts at the center of one circle and ends at the center of a tangent circle

A. is less than the length of the longest radius.B. bends 90° at the point of tangency.C. can’t be drawn as a straight line.D. intersects both circles at the point of tangency.

2. When a circle is drawn through each vertex of a right triangle, the triangle’s hypotenuse will be equal to

A. the diameter of the circle.B. the sum of the lengths of the triangle’s other two sides.C. the circle’s circumference.D. three times the radius of the circle.

3. You have only a pencil, a drafting scale, and a 30°-60° triangle. Given a straight line, you could most easilyuse this equipment to do which of the following?

A. Construct an “S” curveB. Divide the line into several equal partsC. Construct parallel tangent arcsD. Inscribe a square

Examination 35

EXAMINATION NUMBER:

05402000Whichever method you use in submitting your exam

answers to the school, you must use the number above.

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4. You have only a pencil, a straightedge, and a compass. Given a straight line, you could most easily use thisequipment to do which of the following?

A. Form a trapezium and measure its interior anglesB. Construct a dodecahedron out of the given lineC. Make a line parallel to the given lineD. Construct and measure an obtuse angle

5. A right angle can be tangent to a circle at no more than

A. one point.B. two points.C. three points.D. four points.

6. Regarding a circumscribed hexagon,

A. the points of tangency occur at the corners.B. the interior angles are each 30°.C. the distance across opposite corners is equal to the sum of any three sides.D. the distance across the flats is equal to the diameter of the circle.

7. One of the steps in constructing a circumscribed decagon would include

A. using two 30°-60° triangles.B. drawing a square around the outside of the polygon.C. using the concentric circle method.D. dividing a circle into 10 equal parts.

8. You have only a pencil, a straightedge, and a compass. Given a straight line, you could most easily use thisequipment to do which of the following?

A. Construct and measure an acute angleB. Construct a perpendicular bisectorC. Construct a digitized arc tangent to the given lineD. Draw a cube and measure its total surface area

9. How many points of tangency are needed to create an S curve?

A. 0 C. 2B. 1 D. 3

10. An ellipse can be observed whenever

A. the major diameter of the shape is tangent to the minor diameter.B. a circle is viewed perpendicular to the line of sight.C. a circle is viewed at an angle.D. the major diameter of the shape is equal to the minor diameter.

11. A hexahedron is made of only six

A. squares. C. angles.B. hexagons. D. triangles.

12. A right pentagonal prism is made of only

A. five surfaces. C. seven angles.B. five rectangles. D. seven pentagons.

36 Examination

13. You have only a pencil, a straightedge, and a compass. Given a straight line, you could most easily use thisequipment to do which of the following?

A. Divide a circle into 12 equal partsB. Create an ellipseC. Construct a pentagonD. Create a circle tangent to the line

14. Which of the following statements is correct concerning a pentagon inscribed in a circle?

A. The circle will touch all five flats.B. The diameter that intersects one of the shape’s corners will intersect the side opposite that corner.C. The circle will be tangent to the pentagon at six or more points.D. The diameter that bisects one of the shape’s sides will also bisect an opposite side.

15. A truncated right triangular prism has

A. three sides. C. five sides.B. four sides. D. six sides.

16. The acute scalene triangle

A. is a type of quadrilateral.B. is a common regular polygon.C. forms either end of a right rectangular prism.D. has three acute angles.

17. Which of the following collections of equipment would you most likely use when constructing a triangle outof three existing, unconnected lines?

A. A pencil, a compass, and a straightedgeB. A pencil and two drafting scalesC. A pencil, a straightedge, and a French curveD. A pencil, a circle template, and a drafting scale

18. The two diagonals connecting opposite corners of a _______ must always be equal.

A. trapezoid C. trapeziumB. square D. rhomboid

19. You’re given two lines that meet to form an acute angle. With a compass and pencil, you make an arc(arc A) across both lines. You make arc A with the needle point of the compass positioned at the lines’intersection. You then make two more arcs—each with that same radius—within the opening between thelines. To make these arcs, you position the compass point where arc A crosses the existing lines on eitherside of the lines’ intersection. Finally, you draw a line from the intersection of these last two arcs to theintersection of the two existing lines. Which of the following most accurately describes what you’ve justdone?

A. You’ve constructed a perpendicular bisector.B. You’ve transferred a given angle to a new location.C. You’ve constructed a line tangent to an arc.D. You’ve bisected an angle.

20. When constructing a circle that touches a line at only one point, you would usually draw

A. an obtuse angle followed by an ogee curve.B. the circle before attempting to draw the tangent lines.C. the circle before attempting to draw any centerlines.D. an acute angle followed by an S curve.

Examination 37