Question

select the group(s) of sides that will form a right triangle.
□8, 6, 17
□5, 12, 13
□7, 15, 16
□16, 30, 34
□5, 13, 18
□6, 8, 10
□3, 4, 5

Explanation:

To determine if a group of sides forms a right triangle, we use the Pythagorean theorem, which states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\) (where \(c\) is the longest side), \(a^{2}+b^{2}=c^{2}\). We will check each group of sides:

Step 1: Check \(8, 6, 17\)

The longest side is \(17\). Calculate \(8^{2}+6^{2}\) and \(17^{2}\):
\(8^{2}=64\), \(6^{2} = 36\), so \(8^{2}+6^{2}=64 + 36=100\)
\(17^{2}=289\)
Since \(100
eq289\), \(8, 6, 17\) does not form a right triangle.

Step 2: Check \(5, 12, 13\)

The longest side is \(13\). Calculate \(5^{2}+12^{2}\) and \(13^{2}\):
\(5^{2}=25\), \(12^{2}=144\), so \(5^{2}+12^{2}=25 + 144 = 169\)
\(13^{2}=169\)
Since \(169 = 169\), \(5, 12, 13\) forms a right triangle.

Step 3: Check \(7, 15, 16\)

The longest side is \(16\). Calculate \(7^{2}+15^{2}\) and \(16^{2}\):
\(7^{2}=49\), \(15^{2}=225\), so \(7^{2}+15^{2}=49+225 = 274\)
\(16^{2}=256\)
Since \(274
eq256\), \(7, 15, 16\) does not form a right triangle.

Step 4: Check \(16, 30, 34\)

The longest side is \(34\). Calculate \(16^{2}+30^{2}\) and \(34^{2}\):
\(16^{2}=256\), \(30^{2}=900\), so \(16^{2}+30^{2}=256 + 900=1156\)
\(34^{2}=1156\)
Since \(1156=1156\), \(16, 30, 34\) forms a right triangle.

Step 5: Check \(5, 13, 18\)

The longest side is \(18\). Calculate \(5^{2}+13^{2}\) and \(18^{2}\):
\(5^{2}=25\), \(13^{2}=169\), so \(5^{2}+13^{2}=25 + 169 = 194\)
\(18^{2}=324\)
Since \(194
eq324\), \(5, 13, 18\) does not form a right triangle.

Step 6: Check \(6, 8, 10\)

The longest side is \(10\). Calculate \(6^{2}+8^{2}\) and \(10^{2}\):
\(6^{2}=36\), \(8^{2}=64\), so \(6^{2}+8^{2}=36 + 64 = 100\)
\(10^{2}=100\)
Since \(100 = 100\), \(6, 8, 10\) forms a right triangle.

Step 7: Check \(3, 4, 5\)

The longest side is \(5\). Calculate \(3^{2}+4^{2}\) and \(5^{2}\):
\(3^{2}=9\), \(4^{2}=16\), so \(3^{2}+4^{2}=9+16 = 25\)
\(5^{2}=25\)
Since \(25 = 25\), \(3, 4, 5\) forms a right triangle.

Answer:

The groups of sides that form a right triangle are:

• \(5, 12, 13\)
• \(16, 30, 34\)
• \(6, 8, 10\)
• \(3, 4, 5\)