Question

1. show whether each triangle in the table is a right triangle.

triangle\tside lengths (cm)
a\t9, 12, 15
b\t7, 8, 11
c\t7, 24, 25
d\t16, 30, 34
e\t9, 11, 14

Explanation:

To determine if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\) (the legs), and \(c\) (the hypotenuse, the longest side), \(a^{2}+b^{2}=c^{2}\) should hold true.

Triangle A: Side lengths \(9\), \(12\), \(15\)

Step 1: Identify the longest side

The longest side is \(15\), so we check if \(9^{2}+ 12^{2}=15^{2}\)

Step 2: Calculate each square

\(9^{2}=81\), \(12^{2} = 144\), \(15^{2}=225\)

Step 3: Check the sum of the squares of the shorter sides

\(81 + 144=225\), and \(15^{2}=225\). So \(9^{2}+12^{2}=15^{2}\)

Triangle B: Side lengths \(7\), \(8\), \(11\)

Step 1: Identify the longest side

The longest side is \(11\), so we check if \(7^{2}+8^{2}=11^{2}\)

Step 2: Calculate each square

\(7^{2} = 49\), \(8^{2}=64\), \(11^{2}=121\)

Step 3: Check the sum of the squares of the shorter sides

\(49+64 = 113
eq121\). So \(7^{2}+8^{2}
eq11^{2}\)

Triangle C: Side lengths \(7\), \(24\), \(25\)

Step 1: Identify the longest side

The longest side is \(25\), so we check if \(7^{2}+24^{2}=25^{2}\)

Step 2: Calculate each square

\(7^{2}=49\), \(24^{2}=576\), \(25^{2} = 625\)

Step 3: Check the sum of the squares of the shorter sides

\(49 + 576=625\), and \(25^{2}=625\). So \(7^{2}+24^{2}=25^{2}\)

Triangle D: Side lengths \(16\), \(30\), \(34\) (assuming the first number is \(16\) as it seems like a typo)

Step 1: Identify the longest side

The longest side is \(34\), so we check if \(16^{2}+30^{2}=34^{2}\)

Step 2: Calculate each square

\(16^{2}=256\), \(30^{2}=900\), \(34^{2}=1156\)

Step 3: Check the sum of the squares of the shorter sides

\(256+900 = 1156\), and \(34^{2}=1156\). So \(16^{2}+30^{2}=34^{2}\)

Triangle E: Side lengths \(9\), \(11\), \(14\)

Step 1: Identify the longest side

The longest side is \(14\), so we check if \(9^{2}+11^{2}=14^{2}\)

Step 2: Calculate each square

\(9^{2}=81\), \(11^{2}=121\), \(14^{2}=196\)

Step 3: Check the sum of the squares of the shorter sides

\(81 + 121=202
eq196\). So \(9^{2}+11^{2}
eq14^{2}\)

Answer:

• Triangle A: Right triangle (since \(9^{2}+12^{2}=15^{2}\))
• Triangle B: Not a right triangle (since \(7^{2}+8^{2}

eq11^{2}\))

• Triangle C: Right triangle (since \(7^{2}+24^{2}=25^{2}\))
• Triangle D: Right triangle (since \(16^{2}+30^{2}=34^{2}\))
• Triangle E: Not a right triangle (since \(9^{2}+11^{2}

eq14^{2}\))