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Chapter 1: Problem 25

Show that the points form the vertices of the indicated polygon. $$\text { Right triangle: }(4,0),(2,1),(-1,-5)$$

Short Answer

Expert verified
Yes, the points (4,0), (2,1), and (-1,-5) form the vertices of a right triangle.

Step by step solution

01

Calculate distances between points

The first step is to calculate the distances between each pair of points using the distance formula: \(d= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Calculate the distance between (4,0) and (2,1): \(d_1 = \sqrt{(2 - 4)^2 + (1 - 0)^2} = \sqrt{5}\). Calculate the distance between (2,1) and (-1,-5): \(d_2 = \sqrt{(-1 - 2)^2 + (-5 - 1)^2} = \sqrt{45}\). And then calculate for (4,0) and (-1,-5): \(d_3 = \sqrt{(-1 - 4)^2 + (-5 - 0)^2}= \sqrt{50}\).
02

Check the Pythagorean theorem

Now, check if the sum of the squares of the two smaller distances equals the square of the longer distance which is a requirement for a right triangle. The two smaller distances are \( \sqrt{5}\) and \( \sqrt{45}\), and the longest is \( \sqrt{50}\). So, \( (\sqrt{5})^2 + (\sqrt{45})^2 = 5 + 45 = 50\) which equals \( (\sqrt{50})^2\). So, the points given do indeed form a right-angle triangle.

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