Pythagorean Theorem

Show your work for each problem.

1.
In a right triangle, which side is always the longest?
1. a)The hypotenuse
2. b)They are all equal
3. c)The longer leg
4. d)The shorter leg
Answer: The hypotenuse
The hypotenuse is always the longest side of a right triangle. It's the side opposite the right angle (90°).
2.
The Pythagorean theorem states that $a^2 + b^2 = c^2$ . What does $c$ represent?
1. a)The area
2. b)One of the legs
3. c)The hypotenuse
4. d)The perimeter
Answer: The hypotenuse
In the Pythagorean theorem $a^2 + b^2 = c^2$ , $c$ represents the hypotenuse (the longest side), while $a$ and $b$ are the two legs.
3.
A right triangle has legs of length 30 and 40. What is the length of the hypotenuse?
Answer: 50
$30^2 + 40^2 = 900 + 1600 = 2500$ . Since $c^2 = 2500$ , $c = \sqrt{2500} = 50$ . This is the 3-4-5 triangle scaled by 10!
4.
Calculate: $5^2 + 12^2 = ?$
Answer: 169
$5^2 + 12^2 = 25 + 144 = 169$ . Notice that $\sqrt{169} = 13$ , so 5-12-13 is a Pythagorean triple!
5.
Find the hypotenuse of a right triangle with legs of length 6 and 8.
Answer: 10
1. What is $6^2$ ? 36
2. What is $8^2$ ? 64
3. What is $36 + 64$ ? 100
4. What is $\sqrt{100}$ ? 10
6.
A right triangle has a hypotenuse of 26 and one leg of 10. Find the other leg.
Answer: 24
1. What is $26^2$ ? 676
2. What is $10^2$ ? 100
3. What is $676 - 100$ ? 576
4. What is $\sqrt{576}$ ? 24
7.
Which set of numbers forms a Pythagorean triple?
1. a)8, 15, 17
2. b)4, 5, 6
3. c)7, 8, 9
4. d)5, 6, 7
Answer: 8, 15, 17
$8^2 + 15^2 = 64 + 225 = 289 = 17^2$ . The 8-15-17 triangle is a Pythagorean triple! The other options don't satisfy $a^2 + b^2 = c^2$ .
8.
A 5-meter ladder leans against a wall with its foot 1.4 meters from the wall. How high up the wall does it reach, in meters?
Answer: 4.8
Using $a^2 + b^2 = c^2$ : $1.4^2 + h^2 = 5^2$ , so $1.96 + h^2 = 25$ , $h^2 = 23.04$ , $h = \sqrt{23.04} = 4.8$ meters.
9.
A ship sails 16 km east, then 12 km north. How far is it from its starting point?
Answer: 20
1. What is $16^2$ ? 256
2. What is $12^2$ ? 144
3. What is $256 + 144$ ? 400
4. What is $\sqrt{400}$ ? 20
10.
A triangle has sides 6, 8, and 11. Is it a right triangle?
1. a)No, because 11 is odd
2. b)Yes, because $6 + 8 > 11$
3. c)Yes, because it has three sides
4. d)No, because $6^2 + 8^2 \neq 11^2$
Answer: No, because $6^2 + 8^2 \neq 11^2$
Check: $6^2 + 8^2 = 36 + 64 = 100$ , but $11^2 = 121$ . Since $100 \neq 121$ , this is NOT a right triangle.
11.
Find the hypotenuse of a right triangle with legs 20 and 21.
Answer: 29
$20^2 + 21^2 = 400 + 441 = 841 = 29^2$ . The hypotenuse is 29. The 20-21-29 triangle is a less famous Pythagorean triple!
12.
A rectangular TV screen is 48 inches wide and 36 inches tall. What is its diagonal size?
Answer: 60
1. What is $48^2$ ? 2304
2. What is $36^2$ ? 1296
3. What is $2304 + 1296$ ? 3600
4. What is $\sqrt{3600}$ ? 60
13.
A right triangle has a hypotenuse of 17 and one leg of 15. Find the other leg.
Answer: 8
1. What is $17^2$ ? 289
2. What is $15^2$ ? 225
3. What is $289 - 225$ ? 64
4. What is $\sqrt{64}$ ? 8
14.
What is $\sqrt{81}$ ?
Answer: 9
$9 \times 9 = 81$ , so $\sqrt{81} = 9$ . Knowing perfect squares helps solve Pythagorean problems faster!
15.
A wooden garden gate is a rectangle 120 cm wide and 50 cm tall. How long, in cm, is the diagonal brace that runs from one corner to the opposite corner?
Answer: 130
1. What is $120^2$ ? 14400
2. What is $50^2$ ? 2500
3. What is $14400 + 2500$ ? 16900
4. What is $\sqrt{16900}$ ? 130