Pythagorean Theorem

Question 1

A rectangular parking lot is 30 m long and 16 m wide. How long is the diagonal line from one corner to the opposite corner?

Answer 1

Use Pythagorean Theorem: c² = 30² + 16² → c = √(900 + 256) = √1156 = 34 m

Question 2

A hiking trail goes 5 km east then 12 km north. How long is a shortcut directly to the lookout point?

Answer 2

c² = 5² + 12² → c = √(25 + 144) = √169 = 13 km

Question 3

A tent rope is tied to the top of a 2.5 m pole and staked 0.7 m away from the base. How long is the rope?

Answer 3

c² = 2.5² + 0.7² → c = √(6.25 + 0.49) = √6.74 ≈ 2.6 m

Question 4

In a right triangle, the hypotenuse is 13 cm and one leg is 5 cm. Find the missing leg.

Answer 4

a² = 13² − 5² → a = √(169 − 25) = √144 = 12 cm

Question 5

A 10 m ladder leans against a wall, base 6 m away. How high up the wall does the ladder reach?

Answer 5

h² = 10² − 6² → h = √(100 − 36) = √64 = 8 m

Question 6

The temperature drops 4°C each night for 5 nights, starting at 10°C. What is the temperature after 5 nights?

Answer 6

10 − 4×5 = 10 − 20 = −10°C

Question 7

A square garden has an area of 225 m². What is the side length?

Answer 7

side = √225 = 15 m

Question 8

A room measures 8 m by 15 m. What is the diagonal length across the room?

Answer 8

c² = 8² + 15² → c = √(64 + 225) = √289 = 17 m

Question 9

A scuba diver starts at 0 m, dives to −18 m, swims up 7 m, then dives 5 m down. What is the final depth?

Answer 9

−18 + 7 − 5 = −16 m

Question 10

Find the area of a trapezoid with bases 6 m and 10 m and height 5 m. Use A = (b₁ + b₂)h ÷ 2.

Answer 10

A = (6 + 10)×5 ÷ 2 = 16×5 ÷ 2 = 80 ÷ 2 = 40 m²

Question 11

What is the formula for the Pythagorean Theorem?

Answer 11

a² + b² = c²

Question 12

In the Pythagorean Theorem, what does c represent?

Answer 12

The hypotenuse (longest side)

Question 13

What are the sides of the smallest Pythagorean triple?

Answer 13

3, 4, 5

Question 14

If a right triangle has legs 3 and 4, what is the hypotenuse?

Answer 14

5

Question 15

If a right triangle has a hypotenuse of 5 and one leg of 4, what is the other leg?

Answer 15

3

Question 16

Which Pythagorean triple comes after 3-4-5 in size?

Answer 16

5, 12, 13

Question 17

What are the numbers in the Pythagorean triple with hypotenuse 13?

Answer 17

5, 12, 13

Question 18

What are the numbers in the Pythagorean triple with hypotenuse 25?

Answer 18

7, 24, 25

Question 19

What are the numbers in the Pythagorean triple with hypotenuse 17?

Answer 19

8, 15, 17

Question 20

What is the hypotenuse of a triangle with legs 7 and 24?

Answer 20

25

Question 21

True or False: 6, 8, 10 is a Pythagorean triple.

Answer 21

True (it’s a multiple of 3-4-5)

Question 22

True or False: 9, 12, 15 is a Pythagorean triple.

Answer 22

True (it’s a multiple of 3-4-5)

Question 23

True or False: 10, 11, 15 is a Pythagorean triple.

Answer 23

False

Question 24

Fill in the blank: a right triangle with sides 9 and 40 has hypotenuse ___.

Answer 24

41

Question 25

Fill in the blank: a right triangle with hypotenuse 41 and one leg 40 has other leg ___.

Answer 25

9

Question 26

What is the general name for sets like 3-4-5 or 8-15-17?

Answer 26

Pythagorean triples

Question 27

If a right triangle’s legs are equal, what kind of triangle is it?

Answer 27

Isosceles right triangle

Question 28

What is the hypotenuse of an isosceles right triangle with legs of length 5?

Answer 28

5√2

Question 29

What is the area formula for a square?

Answer 29

A = s² (side × side)

Question 30

If a square has side length 12, what is its area?

Answer 30

144

Question 31

If a square has area 169, what is its side length?

Answer 31

13

Question 32

If a square’s diagonal is 10, what is its area?

Answer 32

50 (because side = 10 ÷ √2, then square it)

Question 33

True or False: The area of a square is always larger than the area of a triangle with the same base and height.

Answer 33

True

Question 34

True or False: The area of a square can be a perfect square number.

Answer 34

True (always!)

Question 35

If the area of a square is 225, what is the perimeter?

Answer 35

60 (side = 15, perimeter = 4×15)

Question 36

If a square’s area is 100, what is the length of its diagonal?

Answer 36

10√2

Question 37

What happens to the area of a square if you double its side length?

Answer 37

It becomes 4 times bigger.

Question 38

Which grows faster when side length increases: area or perimeter?

Answer 38

Area (because it grows by side²)

Question 39

What is the unit for area of a square if sides are in cm?

Answer 39

cm²