High School

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1. What is the formula for the Pythagorean theorem?
A. \(a^2 + b^2 = c^2\)
B. \(a^2 - b^2 = c^2\)
C. \(b^2 + c^2 = a^2\)
D. \(c^2 + a^2 = b^2\)

2. If the lengths of two sides of a right triangle are 12 cm and 5 cm respectively, what is the length of the third side?
A. 10 cm
B. 11 cm
C. 12 cm
D. 13 cm

3. What part of a right triangle is described as the longest side and where can it be found in relation to the 90° angle?
A. adjacent
B. hypotenuse
C. opposite
D. right angle

4. What side of a right triangle is always across from the angle to which you are referring?
A. adjacent
B. hypotenuse
C. opposite
D. right angle

5. What side of a right triangle is next to the angle to which you are referring?
A. adjacent
B. hypotenuse
C. opposite
D. right angle

6. Do the segment lengths 15, 12, and 9 form a right triangle?
A. Yes
B. No
C. Maybe
D. Pythagoras

7. Which set of sides would make a right triangle?
A. 4, 5, 6
B. 8, 10, 12
C. 5, 12, 13
D. 5, 10, 12

8. Which equation can be used to solve for "x"?
```
x
|
| 21
|\
| \
| \
| \
| \
|______\
15
```

9. Determine the length of the missing side.
```
|
| 15
|\
| \
| \
| \
| \
|______\
8
```
A. 7
B. 8
C. 17
D. 23

10. Which equation can be used to solve the word problem: The diagonal of a TV is 40 inches and the height is 30 inches. How wide is the TV?

11. You need a ladder that will reach up a 25-foot tall house when placed 10 feet away from the house. How tall does the ladder need to be?
A. 25 feet
B. 26 feet
C. 27 feet
D. 28 feet

12. The slide at the playground has a height of 6 feet. The base of the slide measured on the ground is 8 feet. What is the length of the slide?
A. 8 feet
B. 5.29 feet
C. 10 feet
D. 14 feet

13. A piece of paper that Brittany has is 11 inches tall and 8 inches wide. She draws a straight line diagonally across the paper. How long is the line she drew?
A. 7.5 in
B. 12.4 in
C. 13.6 in
D. 14.3 in

Answer :

Let's go through each part of the student's question step by step:


  1. What is the formula for the Pythagorean theorem?


    • The Pythagorean theorem is a fundamental principle in geometry that relates the three sides of a right triangle. The formula is [tex]a^2 + b^2 = c^2[/tex], where [tex]c[/tex] is the hypotenuse, the side opposite the right angle, and [tex]a[/tex] and [tex]b[/tex] are the other two sides.

    • Answer: A. [tex]a^2 + b^2 = c^2[/tex]



  2. If the lengths of two sides of a right triangle are 12 cm and 5 cm respectively, what is the length of the third side?


    • If these are the two shorter sides, the hypotenuse can be calculated as follows: [tex]\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13[/tex] cm.

    • Answer: D. 13 cm



  3. What part of a right triangle is being described as the longest side, and where can it be found in relation to the 90° angle?


    • The longest side of the right triangle is called the hypotenuse, and it is always found opposite the 90° angle.

    • Answer: B. hypotenuse



  4. What side of a right triangle is always across from the angle to which you are referring?


    • This side is called the opposite side.

    • Answer: C. opposite



  5. What side of a right triangle is next to the angle to which you are referring?


    • This side is called the adjacent side.

    • Answer: A. adjacent



  6. Do the segment lengths 15, 12, and 9 form a right triangle?


    • To verify, check if [tex]15^2 = 12^2 + 9^2[/tex]. Calculate: [tex]225 \neq 144 + 81 = 225[/tex], so they do not form a right triangle.

    • Answer: B. No



  7. Which set of sides would make a right triangle?


    • We need to check which set satisfies [tex]a^2 + b^2 = c^2[/tex]. Only 5, 12, 13 satisfies this: [tex]5^2 + 12^2 = 13^2[/tex]. Calculate: [tex]25 + 144 = 169[/tex].

    • Answer: C. 5, 12, 13



  8. Which equation can be used to solve for "x"?


    • This question is missing some information or choices. Hence, it can't be effectively answered without additional context.



  9. Determine the length of the missing side.


    • Given one side as 15 and the other as 8, the missing side can be calculated using [tex]\sqrt{15^2 - 8^2} = \sqrt{225 - 64} = \sqrt{161}[/tex]. However, the possible answer is not exact without approximation.

    • Answer: Not explicitly given without approximating [tex]\sqrt{161}[/tex].



  10. Which equation can be used to solve the word problem: the diagonal of a t.v. is 40 inches and the height is 30. How wide is the t.v.?




  • Apply the Pythagorean theorem: [tex]30^2 + \text{width}^2 = 40^2[/tex]. Solve for the width.

  • Similar information or context might be required in the form of possible answers.



  1. You need a ladder that will reach up a 25 foot tall house when placed 10 feet away from the house. How tall does the ladder need to be?



  • This uses the Pythagorean theorem: [tex]25^2 + 10^2 = \text{ladder length}^2[/tex]. Calculate: [tex]\sqrt{625 + 100} = \sqrt{725} \approx 26.9[/tex].

  • Answer: B. 26 feet (approximation)



  1. The slide at the playground has a height of 6 feet. The base of the slide measured on the ground is 8 feet. What is the length of the slide?



  • Using the Pythagorean theorem: [tex]\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10[/tex] feet.

  • Answer: C. 10 feet



  1. A piece of paper that Brittany has is 11 inches tall and 8 inches wide. She draws a straight line diagonally across the paper. How long is the line she drew?



  • Again, the Pythagorean theorem: [tex]\sqrt{11^2 + 8^2} = \sqrt{121 + 64} = \sqrt{185} \approx 13.6[/tex] inches.

  • Answer: C. 13.6 in


These problems are all examples of applications of the Pythagorean theorem to solve for side lengths in right triangles.

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