Pre-AP Geometry Worksheet 5.1

Pre-AP Geometry Name: ___________________ Worksheet 5.1: Pythagorean Theorem and its Converse Date: _______ Period: ____ Find the length of each side. Leave answers in simplest radical form on #1 - 4. Round to the nearer

tenth when necessary on #5 – 9. Show work. 1) x = _______ 2) x = ________ 3) x = ______

5) The roof shown in the diagram above is shown from the front of the

4) x = _______ house. The slope of the roof is 125 . The height of the roof is 15 feet.

What is the length from gutter to peak of the roof? 6) An oilrig in the Gulf rises 48 meters above the surface of the sea and sits on a platform anchored to the bottom of the Gulf. In a particularly severe storm, the rig capsized. A nearby TV helicopter filmed its top entering the water 100 meters from the point where the base of the rig was before capsizing. How tall is the platform on which the oilrig sits? Justify your answer.

10 x 13

30

8

13 x x x

11

12

20

x

x

Pre-AP Geometry Worksheet 5.1

7) A small commuter airline flies to three cities whose locations form the vertices of a right triangle. The total flight distance (from city A to city B to city C and back to City A) is 72 miles. It is 30 miles between the two cities that are furthest apart. Find the other two distances between cities. 8) What is the area of the largest square in the diagram? A 5 units2 B 9 units2 C 16 units2 D 25 units2 9) A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the

top of the ladder slips down 4 feet, how many feet will the bottom slide out? No, the answer is not 4 feet. (Hint: draw two right triangles.)

City B

City A City C

4. Justify Mathematical Arguments (1)(G) Write a paragraph proof to prove the Pythagorean Theorem.

Given: △ABC is a right triangle.

Prove: a2 + b2 = c2

5. Apply Mathematics (1)(A) A painter leans a 15-ft ladder against a house. The base of the ladder is 5 ft from the house. To the nearest tenth of a foot, how high on the house does the ladder reach?

Find the value of x. Express your answer in simplest radical form.

6.

x19

16

7.

x

x

8

8.

x

5

10

Is each triangle a right triangle? Explain.

9. 10. 11.

12. Apply Mathematics (1)(A) You want to embroider a square design. You have an embroidery hoop with a 6-in. diameter. Find the largest value of x so that the entire square will fit in the hoop. Round to the nearest tenth.

13. In parallelogram RSTW, RS = 7, ST = 24, and RT = 25. Is RSTW a rectangle? Explain.

14. Analyze Mathematical Relationships (1)(F) △ABC has vertices A(-6, 6), B(-2, -4), and C(1, -1). Verify that AC # BC without using the Slope Formula.

Find the value of x. If your answer is not an integer, express it in simplest radical form.

15. 16. 17.

For each pair of numbers, find a third whole number such that the three numbers form a Pythagorean triple.

18. 20, 21 19. 14, 48 20. 13, 85 21. 12, 37

Proof

B

D

C

ca

b

e

d

A

2820

19

258

24

6533

56

x

x

x26 26

48

x

4 16

4 V5x

3

3

2

428 Lesson 10-1 The Pythagorean Theorem and Its Converse

22. A walkway forms one diagonal of a square playground. The walkway is 24 m long. To the nearest meter, how long is a side of the playground?

The lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse.

23. 13, 2, 3 24. 30, 40, 50 25. 111, 17, 4

26. Apply Mathematics (1)(A) The Hubble Space Telescope orbits 600 km above Earth’s surface. Earth’s radius is about 6370 km. Use the Pythagorean Theorem to find the distance x from the telescope to Earth’s horizon. Round your answer to the nearest ten kilometers. (Diagram is not to scale.)

27. Use the plan and write a proof of Theorem 10-2 (Converse of the Pythagorean Theorem).

Given: △ABC with sides of length a, b, and c, where a2 + b2 = c 2

Prove: △ABC is a right triangle.

Plan: Draw a right triangle (not △ABC) with legs of lengths a and b. Label the hypotenuse x. By the Pythagorean Theorem, a2 + b2 = x 2. Use substitution to compare the lengths of the sides of your triangle and △ABC. Then prove the triangles congruent.

28. Use the plan and write a proof of Theorem 10-3.

Given: △ABC with sides of length a, b, and c, where c 2 7 a2 + b2

Prove: △ABC is an obtuse triangle.

Plan: Draw a right triangle (not △ABC) with legs of lengths a and b. Label the hypotenuse x. By the Pythagorean Theorem, a2 + b2 = x 2. Use substitution to compare lengths c and x. Then use the Converse of the Hinge Theorem to compare ∠C to the right angle.

29. Prove Theorem 10-4.

Given: △ABC with sides of length a, b, and c, where c is the length of the triangle’s longest side and c 2 6 a2 + b2

Prove: △ABC is an acute triangle.

STEM

6370 km

600 kmx

Proof

A

c

b

a

C

B

ProofB

AC b

ca

Proof

C c

a b

A

B

TEXAS Test Practice

30. A 16-ft ladder leans against a building, as shown. To the nearest foot, how far is the base of the ladder from the building?

31. What is the measure of the complement of a 67° angle?

32. The measure of the vertex angle of an isosceles triangle is 58. What is the measure of one of the base angles?

Ladder

15.5 ft

429PearsonTEXAS.com

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