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Chapter 1: Problem 27

Show that the points form the vertices of the indicated polygon. $$\text { Isosceles triangle: }(1,-3),(3,2),(-2,4)$$

Short Answer

Expert verified
Yes, the points (1,-3), (3,2), and (-2,4) form the vertices of an isosceles triangle as two sides are equal in length.

Step by step solution

01

Finding the Distances

Find the distances between the points using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Let's denote the points as follows to make our calculations easier: \(A(1, -3)\), \(B(3, 2)\), \(C(-2, 4)\). Calculate the lengths of AB, BC, and AC.
02

Calculating AB

The distance between points A and B is computed as follows: \(AB = \sqrt{(3 - 1)^2 + (2 - (-3))^2} = \sqrt{4 + 25} = \sqrt{29}\)
03

Calculating BC

Next, find the distance between B and C: \(BC = \sqrt{(-2 - 3)^2 + (4 - 2)^2} = \sqrt{25 + 4} = \sqrt{29}\)
04

Calculating AC

Lastly, find the distance between A and C: \(AC = \sqrt{(-2 - 1)^2 + (4 - (-3))^2} = \sqrt{9 + 49} = \sqrt{58}\)
05

Analyzing the Results

Checking the distances, AB = BC = \(\sqrt{29}\) which are equal and AC = \(\sqrt{58}\). Since two of the sides are equal in length, this indicates that the points A, B, and C do indeed form the vertices of an isosceles triangle.

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