Question

is the length of the unknown leg to the nearest tenth?
a 2.4
b 5.1
c 12.5
d 18.9
35 based on the converse of the pythagorean theorem, which of the following sets of side lengths is not a right triangle?
a $6^2 + 8^2 = 10^2$
b $10^2 + 24^2 = 26^2$
c $2^2 + 3^2 = 4^2$
d $3^2 + 4^2 = 5^2$

Explanation:

First Question (Unknown Leg Length)

Step1: Identify the theorem

We use the Pythagorean theorem for a right triangle, which is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a,b\) are the legs. Here, \(c = 16\), \(b=10\), and we need to find \(a\).

Step2: Rearrange the formula

Rearrange the Pythagorean theorem to solve for \(a\): \(a=\sqrt{c^{2}-b^{2}}\)

Step3: Substitute the values

Substitute \(c = 16\) and \(b = 10\) into the formula: \(a=\sqrt{16^{2}-10^{2}}=\sqrt{256 - 100}=\sqrt{156}\)

Step4: Calculate the square root

Calculate \(\sqrt{156}\approx12.5\) (to the nearest tenth)

The converse of the Pythagorean theorem states that if \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the longest side), then the triangle is a right triangle. We check each option:

• Option A: \(6^{2}+8^{2}=36 + 64=100\) and \(10^{2}=100\), so \(6^{2}+8^{2}=10^{2}\), it is a right triangle.
• Option B: \(10^{2}+24^{2}=100 + 576 = 676\) and \(26^{2}=676\), so \(10^{2}+24^{2}=26^{2}\), it is a right triangle.
• Option C: \(2^{2}+3^{2}=4 + 9 = 13\) and \(4^{2}=16\), since \(13

eq16\), \(2^{2}+3^{2}
eq4^{2}\), it is not a right triangle.

• Option D: \(3^{2}+4^{2}=9+16 = 25\) and \(5^{2}=25\), so \(3^{2}+4^{2}=5^{2}\), it is a right triangle.

Answer:

C. 12.5

Question 35 (Non - right triangle using converse of Pythagorean theorem)