20. Suppose a triangle has sides of length a, b, and c satisfying the equation a? + b? = c?. Show that this triangle is a right triangle.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 45E
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### Problem 20: Identifying a Right Triangle

**Question:**

Suppose a triangle has sides of length \( a \), \( b \), and \( c \) satisfying the equation 

\[ a^2 + b^2 = c^2. \]

Show that this triangle is a right triangle.

**Solution:**

To determine if the given triangle is a right triangle, we will use the Pythagorean theorem. According to the Pythagorean theorem, a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse) is a right triangle if and only if 

\[ a^2 + b^2 = c^2. \]

In this problem, we are given that the sides \( a \), \( b \), and \( c \) satisfy this exact equation. Thus, by the Pythagorean theorem, the given triangle must be a right triangle.

Conclusion: The given triangle is a right triangle because its sides satisfy the Pythagorean equation.

---

This concludes that whenever the equation \( a^2 + b^2 = c^2 \) holds true for the sides of a triangle, the triangle in question is definitively a right triangle.
Transcribed Image Text:--- ### Problem 20: Identifying a Right Triangle **Question:** Suppose a triangle has sides of length \( a \), \( b \), and \( c \) satisfying the equation \[ a^2 + b^2 = c^2. \] Show that this triangle is a right triangle. **Solution:** To determine if the given triangle is a right triangle, we will use the Pythagorean theorem. According to the Pythagorean theorem, a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse) is a right triangle if and only if \[ a^2 + b^2 = c^2. \] In this problem, we are given that the sides \( a \), \( b \), and \( c \) satisfy this exact equation. Thus, by the Pythagorean theorem, the given triangle must be a right triangle. Conclusion: The given triangle is a right triangle because its sides satisfy the Pythagorean equation. --- This concludes that whenever the equation \( a^2 + b^2 = c^2 \) holds true for the sides of a triangle, the triangle in question is definitively a right triangle.
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