Question

jerry drew a figure on the board claiming that it was a right triangle. what side lengths of the figure would show that jerry is correct?
A. 17,18,19
B. 61,90,109
C. 39,52,65
D. 4,5,6

Answer 1

To determine which set of side lengths forms a right triangle, we can use the Pythagorean theorem, which states that for a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the following relationship must hold:

\[
a^2 + b^2 = c^2
\]

Let's evaluate each option:

A. **17, 18, 19**
- Longest side \(c = 19\)
- \(17^2 + 18^2 = 289 + 324 = 613\)
- \(19^2 = 361\)
- \(613 \neq 361\) (not a right triangle)

B. **61, 90, 109**
- Longest side \(c = 109\)
- \(61^2 + 90^2 = 3721 + 8100 = 11821\)
- \(109^2 = 11881\)
- \(11821 \neq 11881\) (not a right triangle)

C. **39, 52, 65**
- Longest side \(c = 65\)
- \(39^2 + 52^2 = 1521 + 2704 = 4225\)
- \(65^2 = 4225\)
- \(4225 = 4225\) (is a right triangle)

D. **4, 5, 6**
- Longest side \(c = 6\)
- \(4^2 + 5^2 = 16 + 25 = 41\)
- \(6^2 = 36\)
- \(41 \neq 36\) (not a right triangle)

Based on this analysis, the only set of side lengths that satisfies the Pythagorean theorem is:

**C. 39, 52, 65**

So, Jerry is correct with these side lengths forming a right triangle.