20014 geometry winter break packet - duval county public...

This practice is a review of topics covered from August to

December.

Topics include:

Parallel lines and transversals

Interior and exterior angle of polygons

Transformations

Triangles

o Building a Fitness Center

o Right triangle Regatta

Quadrilaterals

o Mr. Quad’s Logo

Similarity

Constructions

2014 Geometry

Winter Break Practice

Determine the best

answer for each question.

Demonstrate your

understanding by showing

all your work.

Use additional paper as

needed.

2014 Geometry Winter Break Packet 2

Parallel Lines and Transversals

The first hill of the Steel Dragon 2000 roller coaster in Nagashima, Japan, drops riders from a

height of 318 ft. A portion of this first hill has been transposed onto a coordinate plane and is

shown to the right.

1. The structure of the supports for the hill

consists of steel beams that run parallel and

perpendicular to one another. The endpoints of

the longer of the two support beams

highlighted in Quadrant I are (0, 150) and (120,

0). If the endpoints of the other highlighted

support beam are (0, 125) and (100, 0), verify

and explain why the two beams are parallel.

2. Determine the equations of the lines containing

the beams from Item 1, and explain how the

equations of the lines can help you determine

that the beams are parallel.

3. The equation of a line containing another

support beam is 𝑦 = 4

5 𝑥 + 150. Determine

whether this beam is parallel or perpendicular

to the other two beams, and explain your

reasoning.

4. A linear portion of the first drop is also highlighted in the photo and has endpoints of

(62, 258) and (110, 132). To the nearest foot, determine the distance between the

endpoints of the linear section of the track. Justify your result by showing your work.

5. A camera is being installed at the midpoint of the linear portion of the track described in

Item 4. Determine the coordinates where this camera should be placed.

6. Explain how you could use the distance formula to verify that the coordinates you

determined for the midpoint are correct.


Interior and exterior angles of polygons

Shirley has been hired to organize this year’s Leap Frog Festival and concert in Municipal Park.

Shirley drew a quick sketch (not to scale) of the park to help plan the two events. In the sketch,

there is a gazebo at G, a concession stand at C, and a bridge crossing the creek at the point

labeled B.

Shirley’s plan is to build a stage along the lake wall. She knows that m∠CGB = 72°,

m∠GCB = 47°, and BG = 100 ft. She hopes that the length CG along the wall is at least 100 feet

long to allow room for the stage and food booths on each side.

1. Is CG ≥ 100 ft? Explain how you know.

The gazebo is in the shape of a regular nonagon with the back side against the wall. Shirley

plans to build a triangular planter in the left corner where the wall creates an exterior angle with

the gazebo.

2. What is the degree measure of that exterior angle of the gazebo? Explain how you know.

To create the course for the Leap Frog Contest, portable fence sections will be placed around the

fish pond to create a regular 15-sided polygon as shown.

3. At what angle measure should the consecutive fence sections be connected to create the

regular 15-sided polygon? Explain how you know.


Transformations

In medieval times, a person was rewarded with a coat of arms in recognition of noble acts. A

herald was commissioned to create the coat of arms. In honor of your noble acts thus far in this

course, you are being rewarded with a coat of arms. Each symbol on the grid below represents a

special meaning in the history of heraldry. The acorn in Quadrant I stands for antiquity and

strength. The mascle in Quadrant II represents the persuasiveness you have exhibited in

justifying your answers. The carpenter’s square in Quadrant III represents your conformity to

the laws of right and equity. Finally, the column in Quadrant IV represents the fortitude and

constancy you’ve shown throughout your work in this course.

1. To create your coat of arms, plot the outline of the shield on the coordinate grid on the

grid to the right.

2. Use the transformation (x, y) → (6x, 6y) on the points (2, 2), (−2, 2), (−2, −1), (2, −1),

and (0, −3).

Use after Activity 5.2.

3. Next, transform the figures from their original positions to their intended positions on the

shield.

a. Reflect the acorn over the x-axis.

b. Rotate the mascle 90° clockwise about the origin.

c. Translate the column using (x, y) → (x − 9, y + 10).

d. Rotate the carpenter’s square 90° counter-clockwise about the origin, then reflect it

over the y-axis.


Triangles

1. Use the given angle measures to put side measures x, y, and z in

numerical order. The triangle is not drawn to scale.

2. Tina plans to create triangular flower bed frames using each of the sets of board lengths

listed below. Will each set of boards create a triangle? Explain why or why not.

a. 1 ft , 2 ft , 3 ft b. 6 ft , 9 ft , 11 ft c. 2 ft , 5 ft , 8 ft

3. Given ∆CAT and ∆DOG with measures as shown (not to scale), which of the following

must be true?

a. CT < DG c. CT > DG

b. CT = DG d. Not enough information

4. The given triangle is not drawn to scale.

State which angle must be the largest and explain how you

know.

5. Alex is designing a triangular-shaped park. The park includes tennis courts, ball fields,

and playground equipment located at the vertices of the triangular plot of land. He wants

to locate a water fountain equidistant from each area of the park. Which point of

concurrency would be most helpful to Alex? Explain your reasoning.

6. Jerome is making a sculpture. He wants to balance a scalene triangle on a pointed base.

Which point of concurrency will be most helpful to Jerome? Explain your reasoning.


Building a Fitness Center

The Booster Club at Euclid High School is building a new Fitness Center/Weight Room. The

Fitness Center will be divided into four triangular regions as shown below. It will have cardio

equipment, free weights, resistance machines and mats for warm-up/cool down. Each type of

equipment will be located in a separate triangular region. The Booster Club would like to make

sure that an equal amount of space is allotted for cardio equipment and free weights.

If E is the midpoint of 𝐷𝐵̅̅ ̅̅ and the walls of the fitness center, 𝐴𝐵̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ are parallel, will the

cardio equipment and the free weights have an equal amount of space?

1. Mark the diagram with the given information and any other important facts.

2. Write a congruence statement for the two triangles you would like to show congruent.

State the congruence method you will use to show the triangles congruent.

3. Justify, using a paragraph proof, why these two triangles are congruent.

4. Write a flow chart proof to support your congruence statement.

5. Write a two column proof to show that E is also the midpoint of 𝐴𝐶̅̅ ̅̅ .

6. If the walls of the fitness center form a rectangle, ∠𝐷𝐸𝐶 is an isosceles triangle and the

measure of ∠EDC is 32°. List the measure of all other angles in the diagram. Justify each

answer.


Right Triangle Regatta

Cole is the finish line judge for the annual summer sailboat race on

Lake Vacation. Before the race he helps the other judges set up the

courses for the race. Each of the courses will be triangular in shape.

Sailors will sail due west from the starting line, make a left turn

around a buoy, creating a right angle, and sail due south to the

finish line as in the diagram to the right.

1. There are three proposed triangular courses for the beginner sailors. The lengths of each

leg of the race and the distance from the starting line to the finish line for each proposed

course are listed below. Which course would be appropriate for the Right Triangle

Regatta? Explain your answer, including reasons for not choosing the other two.

a. 20 miles, 26 miles, 34 miles

b. 16 miles, 30 miles, 34 miles

c. 15 miles, 31 miles, 34 miles

The course for the advanced sailors has already been decided. From

the starting line, the boats will travel due west for 9 miles and then

turn 90° around a marker buoy to travel due south 12 miles to the

finish line. There is a 2nd buoy located between the starting line and

the finish line as shown in the diagram to the right.

2. Cole must travel from the starting line to the finish line before the beginning of the race.

What is the shortest distance from Start to Finish? Show the calculations that lead to your

answer.

3. There will be a judge positioned at each buoy and at the starting line and finish line.

a. How far must a judge travel from the starting line to the 2nd buoy not on the race

course? Show the calculations that lead to your answer.

b. Once the judge arrives at the 2nd buoy, how far will he be from the finish line? Show

the calculations that lead to your answer.

4. What will be the distance between the two judges stationed on the two buoys?

Show the calculations that lead to your answer.


Quadrilaterals

1. If ABCD is a parallelogram, list everything you

know about the sides and angles in the figure.

2. Which of the following conditions are sufficient,

individually, to guarantee that ABCD is a

parallelogram?

A. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ ∥ 𝐴𝐷̅̅ ̅̅

B. 𝐵𝐷̅̅ ̅̅ ⊥ 𝐴𝐶̅̅ ̅̅

C. 𝐵𝐷̅̅ ̅̅ and 𝐴𝐶̅̅ ̅̅ bisect each other

D. 𝐵𝐸̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅ and 𝐴𝐸̅̅ ̅̅ ≅ 𝐸𝐷̅̅ ̅̅

E. ∠𝐵𝐴𝐷 ≅ ∠𝐵𝐶𝐷 and ∠𝐴𝐵𝐶 ≅ ∠𝐴𝐷𝐶

F. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅

3. ABCD is a kite, and EFGH is an isosceles trapezoid with 𝐹𝐺̅̅ ̅̅ ∥ 𝐸𝐻̅̅ ̅̅ . Find the value of

each variable, and support each answer by stating a definition or theorem.

4. PQRS is a kite, with 𝑄𝑅̅̅ ̅̅ = 𝑅𝑆̅̅̅̅ = 𝑎, 𝑤ℎ𝑒𝑟𝑒 0 < 𝑎 < 5. a. Give the coordinates of Q and S in terms of a.

b. Prove that 𝑃𝑅̅̅ ̅̅ ⊥ 𝑄𝑆.̅̅̅̅̅


Mr. Quad’s Logo

Thurston B. Quad, III acquired a new company in a hostile takeover. This new company, Math

Manipulatives, needed a new logo. Mr. Quad’s wife teaches geometry, and her class submitted

the following instructions for the design of the logo.

Begin with an isosceles trapezoid and locate the midpoint of each side.

Use these midpoints as the vertices of a new quadrilateral to be formed inside the

first quadrilateral.

Locate the midpoint of each side of the second quadrilateral, and use these

midpoints as vertices to form a third quadrilateral.

Repeat this process with each new quadrilateral until the newest quadrilateral is

too small to be seen.

Suppose you were one of Mr. Quad’s employees, and he assigned you the task of investigating

this design proposal. Write a report to be sent to Mr. Quad. In your report, you should include:

1. A reproduction of this design on a piece of graph paper. You must show at least six

quadrilaterals, including the first.

2. How you know that your first quadrilateral is an isosceles trapezoid.

3. The best name (trapezoid, parallelogram, rectangle, rhombus or square) for each

subsequent quadrilateral and a convincing argument that supports the name you chose.

4. A description of any patterns in the sequence of the shapes that you may find.

5. Definitions, postulates, and theorems from geometry to support your claims.


Similarity

Dan is designing a garden chair. The diagram below shows a side view of the chair when it is set

up for use. 𝑨𝑩̅̅ ̅̅ and 𝑪𝑫̅̅ ̅̅ represent two lengths of wood hinged together at X. 𝑩𝑫̅̅̅̅̅ is horizontal.

A is vertically above D, and C is vertically above B.

1. Calculate the angle between the two lengths of wood, ∠BXC.

2. Use the Pythagorean Theorem to calculate the length of CD.

Show your work.

3. Show that triangles AXD and BXC are similar.

4. Use the fact that triangles AXD and BXC are similar to calculate CX, the distance from

the top of the seat to the hinge.

Show how you figured it out.


Constructions

1. Use construction tools to construct a square using the given perimeter.

2. Point R is the midpoint of line segment PQ. Compare the lengths of 𝑃𝑅̅̅ ̅̅ and 𝑅𝑄̅̅ ̅̅ .

Explain your reasoning.

3. Use construction tools to construct an isosceles triangle that is not an equilateral triangle

with the side length shown.

4. a) Use construction tools to bisect segment AB. Label the midpoint C.

b) Use construction tools to bisect the length of line segment AC. Label the midpoint D.

c) Describe the relationship between the length of line segment AB and the length of line

segment AD. Explain you reasoning.


5. Construct triangle JKL using triangle FGH as a counterexample to show that Angle-

Side-Side is not a valid triangle congruence theorem. Explain your answer.

6. Complete the construction. Then, use the diagram to complete the proof.

a) Complete the following steps:

Draw a line from point A to a point D.

Draw a line from point B to a point C.

Label the intersection of 𝐴𝐷̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ as point G.

Construct the angle bisector of ∠𝐴𝐺𝐵.

Label the intersection of the angle bisector with 𝐴𝐵̅̅ ̅̅ as point E.

Label the intersection of the angle bisector with 𝐶𝐷̅̅ ̅̅ as point F.

b)

Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅

𝐸𝐹̅̅ ̅̅ bisects ∠𝐴𝐺𝐵

Prove: Point G is the midpoint of 𝐸𝐹̅̅ ̅̅

20014 geometry winter break packet - duval county public...

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