Question

sally wants to build an acute triangle. which of these sets of sides could she use? (60, 39, 61) (45, 53, 28) (7, 24, 25) (17, 8, 15)

Explanation:

Step1: Recall acute triangle condition

For a triangle with sides \(a \leq b \leq c\), it is acute if \(a^2 + b^2 > c^2\), right if \(a^2 + b^2 = c^2\), and obtuse if \(a^2 + b^2 < c^2\).

Step2: Check set (60, 39, 61)

Sort: \(39, 60, 61\). Calculate \(39^2 + 60^2 = 1521 + 3600 = 5121\), \(61^2 = 3721\). Wait, no, \(61^2 = 3721\)? Wait, \(60^2=3600\), \(39^2 = 1521\), sum is \(3600 + 1521 = 5121\). \(61^2 = 3721\)? No, \(61\times61 = 3721\)? Wait, no, \(60^2=3600\), \(61^2=3721\), \(39^2=1521\). Wait, \(39^2 + 60^2 = 1521 + 3600 = 5121\), which is greater than \(61^2 = 3721\)? Wait, no, I must have sorted wrong. The largest side is 61, so \(a=39\), \(b=60\), \(c=61\). So \(39^2 + 60^2 = 1521 + 3600 = 5121\), \(61^2 = 3721\)? Wait, no, \(61^2 = (60 + 1)^2 = 60^2 + 2\times60\times1 + 1 = 3600 + 120 + 1 = 3721\). Wait, 5121 > 3721, so \(39^2 + 60^2 > 61^2\), so acute? Wait, no, maybe I mixed up. Wait, no, the largest side is 61, so \(a=39\), \(b=60\), \(c=61\). So \(a^2 + b^2 = 39^2 + 60^2 = 1521 + 3600 = 5121\), \(c^2 = 61^2 = 3721\). Wait, that can't be, 5121 > 3721, so it's acute? Wait, no, maybe I sorted incorrectly. Wait, 60 is longer than 39, 61 is longest. Wait, maybe I made a mistake in the formula. Wait, no, the formula is for \(a \leq b \leq c\), so \(a^2 + b^2 > c^2\) for acute. Wait, but 39 and 60 are smaller than 61? No, 60 is less than 61, 39 is less than 60. So \(39^2 + 60^2 = 1521 + 3600 = 5121\), \(61^2 = 3721\). So 5121 > 3721, so it's acute.

Step3: Check (45, 53, 28)

Sort: \(28, 45, 53\). \(28^2 + 45^2 = 784 + 2025 = 2809\), \(53^2 = 2809\). So \(28^2 + 45^2 = 53^2\), so it's a right triangle, not acute.

Step4: Check (7, 24, 25)

Sort: \(7, 24, 25\). \(7^2 + 24^2 = 49 + 576 = 625\), \(25^2 = 625\). So it's a right triangle, not acute.

Step5: Check (17, 8, 15)

Sort: \(8, 15, 17\). \(8^2 + 15^2 = 64 + 225 = 289\), \(17^2 = 289\). So it's a right triangle, not acute.

Wait, but earlier for (60,39,61), I think I messed up the largest side. Wait, 61 is the largest, so \(39^2 + 60^2 = 1521 + 3600 = 5121\), \(61^2 = 3721\). Wait, 5121 > 3721, so \(a^2 + b^2 > c^2\), so acute. The other sets: (45,53,28) sorted is 28,45,53. \(28^2 + 45^2 = 784 + 2025 = 2809 = 53^2\) (right). (7,24,25): \(7^2 +24^2=49+576=625=25^2\) (right). (17,8,15): \(8^2 +15^2=64+225=289=17^2\) (right). So only (60,39,61) is acute? Wait, but wait, 60, 39, 61: let's recalculate \(60^2 + 39^2 = 3600 + 1521 = 5121\), \(61^2 = 3721\). Wait, that's correct. So \(5121 > 3721\), so acute.

Answer:

(60, 39, 61)