Question

if a triangle has sides of lengths 6, 8, and 10, does it satisfy the pythagorean theorem?
a. cannot be determined
b. yes
c. no
d. sometimes

in a right - triangle, if (a = 7) and (c = 25), what is (b)?
a. 20
b. 24
c. 18
d. 23

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the longest side of a right - triangle and \(a\) and \(b\) are the other two sides.

Step2: Check if the sides 6, 8, 10 satisfy the theorem

Let \(a = 6\), \(b = 8\), and \(c = 10\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}=6^{2}=36\), \(b^{2}=8^{2}=64\), so \(a^{2}+b^{2}=36 + 64=100\). And \(c^{2}=10^{2}=100\). Since \(a^{2}+b^{2}=c^{2}\), the triangle satisfies the Pythagorean Theorem.

Step3: For the right - triangle with \(a = 7\) and \(c = 25\), use the Pythagorean Theorem

The Pythagorean Theorem is \(a^{2}+b^{2}=c^{2}\), we can rewrite it to solve for \(b\) as \(b=\sqrt{c^{2}-a^{2}}\) (assuming \(c\) is the hypotenuse). Substitute \(a = 7\) and \(c = 25\) into the formula: \(b=\sqrt{25^{2}-7^{2}}=\sqrt{(25 + 7)(25 - 7)}=\sqrt{32\times18}=\sqrt{576}=24\).

Answer:

1. b. Yes
2. b. 24