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\t10, 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.

Step 1: Analyze Triangle A (9, 12, 15)

• Identify the longest side: \(c = 15\), and the other two sides \(a = 9\), \(b = 12\).
• Calculate \(a^{2}+b^{2}\): \(9^{2}+12^{2}=81 + 144=225\)
• Calculate \(c^{2}\): \(15^{2}=225\)
• Since \(9^{2}+12^{2}=15^{2}\), Triangle A is a right triangle.

Step 2: Analyze Triangle B (7, 8, 11)

• Longest side \(c = 11\), \(a = 7\), \(b = 8\)
• Calculate \(a^{2}+b^{2}\): \(7^{2}+8^{2}=49+64 = 113\)
• Calculate \(c^{2}\): \(11^{2}=121\)
• Since \(7^{2}+8^{2}

eq11^{2}\) (\(113
eq121\)), Triangle B is not a right triangle.

Step 3: Analyze Triangle C (7, 24, 25)

• Longest side \(c = 25\), \(a = 7\), \(b = 24\)
• Calculate \(a^{2}+b^{2}\): \(7^{2}+24^{2}=49 + 576=625\)
• Calculate \(c^{2}\): \(25^{2}=625\)
• Since \(7^{2}+24^{2}=25^{2}\), Triangle C is a right triangle.

Step 4: Analyze Triangle D (16, 30, 34)

• Longest side \(c = 34\), \(a = 16\), \(b = 30\)
• Calculate \(a^{2}+b^{2}\): \(16^{2}+30^{2}=256+900 = 1156\)
• Calculate \(c^{2}\): \(34^{2}=1156\)
• Since \(16^{2}+30^{2}=34^{2}\), Triangle D is a right triangle.

Step 5: Analyze Triangle E (10, 11, 14)

• Longest side \(c = 14\), \(a = 10\), \(b = 11\)
• Calculate \(a^{2}+b^{2}\): \(10^{2}+11^{2}=100 + 121=221\)
• Calculate \(c^{2}\): \(14^{2}=196\)
• Since \(10^{2}+11^{2}

eq14^{2}\) (\(221
eq196\)), Triangle E is not a right triangle.

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 \(10^{2}+11^{2}

eq14^{2}\))