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Chapter 1: Problem 79

The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle. $$(-4,5),(6,1), \text { and }(-8,-5)$$

Short Answer

Expert verified
The points form a right triangle.

Step by step solution

01

- Find Distance Between Points

Use the distance formula \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) to find the lengths of the sides of the triangle formed by the points (-4,5), (6,1), and (-8,-5).
02

- Calculate Distance AB

Let points A, B, and C be (-4,5), (6,1), and (-8,-5) respectively. Find the distance between points A and B: \[ AB = \sqrt{(6 - (-4))^2 + (1 - 5)^2} = \sqrt{(6 + 4)^2 + (-4)^2} = \sqrt{10^2 + 4^2} = \sqrt{100 + 16} = \sqrt{116} = 2 \sqrt{29} \]
03

- Calculate Distance BC

Find the distance between points B and C: \[ BC = \sqrt{(-8 - 6)^2 + (-5 - 1)^2} = \sqrt{(-14)^2 + (-6)^2} = \sqrt{196 + 36} = \sqrt{232} = 2 \sqrt{58} \]
04

- Calculate Distance CA

Find the distance between points C and A: \[ CA = \sqrt{(-8 - (-4))^2 + (-5 - 5)^2} = \sqrt{(-8 + 4)^2 + (-10)^2} = \sqrt{(-4)^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116} = 2 \sqrt{29} \]
05

- Determine if Right Triangle

Check if the sum of the squares of two sides equals the square of the third side. Compare: \[ (2 \sqrt{29})^2 + (2 \sqrt{29})^2 = (2 \sqrt{58})^2 \] \[ 4 \times 29 + 4 \times 29 = 4 \times 58 \] \[ 116 + 116 = 232 \] The sum is equal, so the points form a right triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Vertices and Right Angles
Vertices are the points where the sides of a triangle intersect. In coordinate geometry, vertices are identified by their coordinates (x, y). To determine if these vertices form a right triangle, first label each vertex clearly (e.g., A, B, and C). For example, given points A(-4,5), B(6,1), and C(-8,-5):
  • Label the points correctly.
  • Calculate the distances AB, BC, and CA using the distance formula.
  • Verify with the converse Pythagorean theorem.
In our specific example, after calculating the distances (\( 2\sqrt{29}, 2\sqrt{58}\) and confirming that \( (2\sqrt{29})^2 + (2\sqrt{29})^2 = (2\sqrt{58})^2 \), we conclude that the points indeed form a right triangle.Understanding vertices and how distances among them relate to right angles can simplify problems involving right triangles.

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