Question

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

triangle	side lengths (cm)
b	7, 8, 11
c	7, 24, 25
d	16, 30, 34
e	10, 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), \(a^{2}+b^{2}=c^{2}\). We will check each triangle by squaring the lengths of the two shorter sides, adding them, and comparing to the square of the longest side.

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

• Identify the longest side: \(c = 15\), shorter 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: Triangle B (7, 8, 11)

• Longest side \(c = 11\), shorter sides \(a = 7\), \(b = 8\).
• 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: Triangle C (7, 24, 25)

• Longest side \(c = 25\), shorter sides \(a = 7\), \(b = 24\).
• 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: Triangle D (16, 30, 34)

• Longest side \(c = 34\), shorter sides \(a = 16\), \(b = 30\).
• 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: Triangle E (10, 11, 14)

• Longest side \(c = 14\), shorter sides \(a = 10\), \(b = 11\).
• Calculate \(a^{2}+b^{2}\): \(10^{2}+11^{2}=100 + 121=221\).
• Calculate \(c^{2}\): \(14^{2}=196\). Wait, no, \(14^{2}=196\)? Wait, \(14\times14 = 196\), but \(10^{2}+11^{2}=221\). Wait, actually, I made a mistake here. Wait, \(14\) is the longest side? Wait, \(10\), \(11\), \(14\): \(14\) is the longest. Wait, \(10^{2}+11^{2}=100 + 121 = 221\), and \(14^{2}=196\)? No, that can't be. Wait, no, \(14^{2}=196\)? Wait, no, \(14\times14 = 196\), but \(221>196\), so \(a^{2}+b^{2}

eq c^{2}\). Wait, actually, I think I mixed up. Wait, the hypotenuse should be the longest side, so if \(c\) is the longest side, then \(a^{2}+b^{2}\) should equal \(c^{2}\). So for triangle E: \(10^{2}+11^{2}=100 + 121 = 221\), \(14^{2}=196\). Since \(221
eq196\), Triangle E is not a right triangle. Wait, but wait, maybe I got the longest side wrong? Wait, \(11<14\), \(10<14\), so 14 is the longest. So yes, \(a^{2}+b^{2}=221\), \(c^{2}=196\), so not equal. So 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}\))