Question

holly and tamar completed the work in the table to determine if a triangle with side lengths of 40, 42, and 58 is a right triangle. hollys work $(40 + 42)^{2}stackrel{?}{=}58^{2}$ $82^{2}stackrel{?}{=}3,364$ $6,724
eq3,364$ the triangle is not a right triangle. tamars work $42^{2}+40^{2}stackrel{?}{=}58^{2}$ $1,764 + 1,600stackrel{?}{=}3,364$ $3,364 = 3,364$ the triangle is a right triangle. which best describes the accuracy of their solutions? neither holly nor tamar is correct. holly is correct. both holly and tamar are correct. tamar is correct.

Explanation:

Step1: Recall the Pythagorean theorem

For a right - triangle with side lengths \(a\), \(b\), and \(c\) (\(c\) is the longest side), the theorem is \(a^{2}+b^{2}=c^{2}\).

Step2: Analyze Holly's work

Holly used \((40 + 42)^{2}\stackrel{?}{=}58^{2}\). The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), not \((a + b)^{2}=c^{2}\). \((a + b)^{2}=a^{2}+2ab + b^{2}\), which is incorrect application of the theorem.

Step3: Analyze Tamar's work

Tamar used \(42^{2}+40^{2}\stackrel{?}{=}58^{2}\). Calculate \(42^{2}=1764\), \(40^{2}=1600\), and \(58^{2}=3364\). Then \(42^{2}+40^{2}=1764 + 1600=3364\) and \(58^{2}=3364\). So \(42^{2}+40^{2}=58^{2}\), which is a correct application of the Pythagorean theorem.

Answer:

Tamar is correct.