Question

find the missing side of the following right triangles using the pythagorean theorem
$a^2 + b^2 = c^2$
show your work to justify your answer!!!
19)
6cm
3cm
x
20)
4cm
3cm
x
21)
14cm
22cm
x

Explanation:

Problem 19:

Step1: Identify sides

Legs: \( a = 6 \, \text{cm} \), \( b = 3 \, \text{cm} \), hypotenuse \( c = x \).

Step2: Apply Pythagorean theorem

\( x^2 = 6^2 + 3^2 \)
\( x^2 = 36 + 9 = 45 \)

Step3: Solve for \( x \)

\( x = \sqrt{45} = 3\sqrt{5} \approx 6.71 \, \text{cm} \)

Problem 20:

Step1: Identify sides

Legs: \( a = 4 \, \text{cm} \), \( b = 7 \, \text{cm} \), hypotenuse \( c = x \).

Step2: Apply Pythagorean theorem

\( x^2 = 4^2 + 7^2 \)
\( x^2 = 16 + 49 = 65 \)

Step3: Solve for \( x \)

\( x = \sqrt{65} \approx 8.06 \, \text{cm} \)

Problem 21:

Step1: Identify sides

Legs: \( a = 14 \, \text{cm} \), \( b = 22 \, \text{cm} \), hypotenuse \( c = x \).

Step2: Apply Pythagorean theorem

\( x^2 = 14^2 + 22^2 \)
\( x^2 = 196 + 484 = 680 \)

Step3: Solve for \( x \)

\( x = \sqrt{680} = 2\sqrt{170} \approx 26.08 \, \text{cm} \)

Answer:

s:

1. \( 3\sqrt{5} \, \text{cm} \) (or \( \approx 6.71 \, \text{cm} \))
2. \( \sqrt{65} \, \text{cm} \) (or \( \approx 8.06 \, \text{cm} \))
3. \( 2\sqrt{170} \, \text{cm} \) (or \( \approx 26.08 \, \text{cm} \))