honors geometry lexington high school 2012 … geometry 2012 final exam lexington high school june...

HONORS GEOMETRY 2012 Final Exam

Lexington High School June 13, 2012

Your Name:

Teacher (circle): Doucette Haupt Kelly Olsen Richardson

Block (circle): A B C E F G H

Directions: See the beginning of each part of the exam for directions.

Timing: 90 minutes expected, 120 minutes maximum.

Only permitted items: pens and pencils, calculators, scrap paper given by your proctor.

your scores

Part A. Numerical answers 6 problems 3 points each /18

Part B. Proofs based on diagrams 2 problems 5 points each /10

Part C. Free response questions 3 problems 5 points each /15

Part D. Proofs of major theorems 3 problems 5 points each /15

TOTAL /58

Part A. Numerical answers Each question asks you to find a length, area, volume, or angle measure. These questions will be

graded right/wrong based solely on your numerical answers. Write your numerical answers in

the answer boxes. Answers may be given either exactly or approximately. Approximate answers

must be accurate to at least 3 decimal places. Some partial credit is available for correct work.

1. In a triangle whose side lengths are 6, 7, and 8, what is the degree measure of the triangle’s

smallest angle?

Answer to problem 1:

The measure of the smallest angle is degrees.

Honors Geometry 2012 Final Exam page 2

2. In the given diagram, O is the center of the circle, L is a

point of tangency, OL = 3, and LI = 5.

Find the area of sector LOK of the circle.


The area of sector LOK is square units.

3. Circles centered at I and T are internally tangent to

each other and tangent to MK at point M. Points T,

I, and M are collinear. If arc MR and arc TA have

equal degree measures, what is the degree measure

of K?


The measure of K is degrees.


4. Lines k and n intersect at point Y forming 55° and 125° angles.

Point U is the reflected image of point F across line k. Point R

is the reflected image of point U across line n. Find the degree

measure of FUR.


The measure of FUR is degrees.

5. Given an equilateral triangle with each side measuring

6 centimeters, a circle is inscribed in the triangle and then a

square is inscribed in the circle. Find the square’s area.


The square’s area is square centimeters.


6. A regular tetrahedron is a polyhedron whose 4 faces are equilateral triangles. If a

regular tetrahedron has edges of length 8 centimeters, what is the tetrahedron’s volume?


The tetrahedron’s volume is cubic centimeters.


Part B. Proofs with diagrams provided This part of the test has 3 questions. In each question you are shown a diagram, given

information, and asked to prove something about the diagram. You must choose exactly 2

of the 3 questions to answer. You may use any proof technique you have learned in the course

(deductive, coordinate, etc.). When writing deductive proofs, you may cite as evidence any

postulates or theorems that preceded the theorem you are proving, in the order of development

used in your class or your textbook.

MANDATORY CHOICE: In this box, circle the numbers of the 2 problems you are choosing.

(If you do not indicate choices, the first 2 problems where you wrote anything will be graded.)

7 8 9

Write complete, fully justified proofs. You may cite as evidence any postulates or theorems you

have studied in this course.

7. LJ and LR are rays contained by plane N. M is a

point not on the plane. The perpendicular from

M to LJ meets LJ at O. The perpendicular from M

to LR meets LR at I.

Prove that if MO = MI then LO = LI.


8. In the given diagram:

Arc MULD is a semi-circle centered at A.

Points H, U, L, and K are collinear.

MH HK and HK KD.

Prove that (HU)2 + (UK)

2 = (HL)

2 + (LK)

2.

9. The diagonals of a parallelogram WIDO are each

subdivided into 6 equal parts, then certain points,

as shown in the diagram, are connected to form a

new quadrilateral BLCK. What kind of special

quadrilateral is BLCK? Prove your answer.


Part C. Free response questions Show and justify your steps as you answer these questions. For full credit, your answers must be

both correct and fully justified. You may cite as evidence any postulates or theorems you have

studied in this course.

10. A regular polygon has angles of 150° and is inscribed in a circle of radius 4. A regular

octagon is inscribed in the same circle. What is the ratio between the n-gon’s area and the

octagon’s area? Justify your answer.

11. Four metal balls (sphere-shaped) fit snugly inside a cylindrical can, as illustrated.

A student claims that two more balls of the same size can be put into the can,

provided all six balls can be melted down and the molten liquid poured into the can.

Is the student correct? Justify your answer.


12. Given: parallelogram THOR with coordinates

as shown in the diagram.

Answer parts a, b and c below.

a. In terms of the variables already

established, what are the coordinates of

point T? Justify your answer.

b. Find the coordinates of the point where diagonals HR and TO intersect.

c. Using coordinates, show that the diagonals HR and TO bisect each other.


Part D. Proofs of major theorems This part of the test has 5 questions, each concerning the proof of a major theorem of geometry.

You must choose exactly 3 of the 5 questions to answer. You may use any proof technique you

have learned in the course (deductive, coordinate, etc.). When writing deductive proofs, you may

cite as evidence any postulates or theorems that preceded the theorem you are proving, in the

order of development used in your class or your textbook.

MANDATORY CHOICE: In this box, circle the numbers of the 3 problems you are choosing.

(If you do not indicate choices, the first 3 problems where you wrote anything will be graded.)

13 14 15 16 17

13. Prove the converse of the Pythagorean Theorem: If the square of one side of a triangle is

equal to the sum of the squares of the other two sides, then the triangle is a right triangle.


14. State and prove the Law of Sines.

15. Prove that in a quadrilateral, if one pair of sides is both equal and parallel, then the

quadrilateral is a parallelogram.


16. Prove that the three angle bisectors of a triangle must be concurrent.


17. If a tangent and a secant are drawn to a circle

from an external point W, then the square of

the tangent is equal to the product of the secant

and its external segment (WK2 = WH WA).

honors geometry lexington high school 2012 … geometry 2012 final exam lexington high school june...

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