Question

which theorem helps in finding the unknown side length of a right triangle? a. binomial theorem b. pythagorean theorem c. mean value theorem d. fundamental theorem of algebra if the hypotenuse of a right triangle is 25 units and one leg is 24 units, what is the length of the other leg? a. 9 units b. 15 units c. 7 units d. 20 units

Explanation:

Step1: Recall triangle - related theorems

The Pythagorean Theorem is used for right - triangles to find unknown side lengths. It states that in a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). The Binomial Theorem is for expanding \((a + b)^n\), the Mean Value Theorem is in calculus for functions, and the Fundamental Theorem of Algebra is about the roots of polynomials.

Step2: Apply the Pythagorean Theorem

Let the hypotenuse \(c = 25\) and one leg \(a = 24\). According to the Pythagorean Theorem \(b^{2}=c^{2}-a^{2}\). Substitute the values: \(b^{2}=25^{2}-24^{2}=(25 + 24)(25 - 24)\) (using the difference - of - squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)). Then \(b^{2}=(49)(1)=49\), so \(b = 7\).

Answer:

1. b. Pythagorean Theorem
2. c. 7 units