Question

use the pythagorean theorem to find the missing side length for each of the right triangles below.
1.
$a =$____
$b =$____
$c =$____
2.
$a =$____
$b =$____
$c =$____
3.
$a =$____
$b =$____
$c =$____
4.
$a =$____
$b =$____
$c =$____
5.
$a =$____
$b =$____
$c =$____
6.
$a =$____
$b =$____
$c =$____

Explanation:

1.

Step1: Label the sides

Let \(a = 3\), \(b=4\). By the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\).

Step2: Calculate \(c\)

\(c^{2}=3^{2}+4^{2}=9 + 16=25\), so \(c=\sqrt{25}=5\).

2.

Step1: Identify the sides

Let \(a = 5\), \(c = 13\). By the Pythagorean theorem \(b^{2}=c^{2}-a^{2}\).

Step2: Calculate \(b\)

\(b^{2}=13^{2}-5^{2}=169 - 25 = 144\), so \(b=\sqrt{144}=12\).

3.

Step1: Label the sides

Let \(b = 7\), \(c = 25\). By the Pythagorean theorem \(a^{2}=c^{2}-b^{2}\).

Step2: Calculate \(a\)

\(a^{2}=25^{2}-7^{2}=625-49 = 576\), so \(a=\sqrt{576}=24\).

4.

Step1: Identify the sides

Let \(c = 17\), \(b = 15\). By the Pythagorean theorem \(a^{2}=c^{2}-b^{2}\).

Step2: Calculate \(a\)

\(a^{2}=17^{2}-15^{2}=289 - 225=64\), so \(a = 8\).

5.

Step1: Label the sides

Let \(a = 40\), \(b = 9\). By the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\).

Step2: Calculate \(c\)

\(c^{2}=40^{2}+9^{2}=1600+81 = 1681\), so \(c=\sqrt{1681}=41\).

6.

Step1: Identify the sides

Let \(a = 60\), \(c = 61\). By the Pythagorean theorem \(b^{2}=c^{2}-a^{2}\).

Step2: Calculate \(b\)

\(b^{2}=61^{2}-60^{2}=3721 - 3600=121\), so \(b=\sqrt{121}=11\).

Answer:

1. \(a = 3\), \(b = 4\), \(c = 5\)
2. \(a = 5\), \(b = 12\), \(c = 13\)
3. \(a = 24\), \(b = 7\), \(c = 25\)
4. \(a = 8\), \(b = 15\), \(c = 17\)
5. \(a = 40\), \(b = 9\), \(c = 41\)
6. \(a = 60\), \(b = 11\), \(c = 61\)