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Chapter 7: Problem 12

a. Plot the points \(A(1,-3), B(3,2),\) and \(C(-2,4)\), then connect the points to form a triangle. (GRAPH CAN NOT COPY) b. Use the distance formula to verify that those points are vertices of an isosceles triangle.

Short Answer

Expert verified
Answer: Yes, the points A(1, -3), B(3, 2), and C(-2, 4) form an isosceles triangle with sides AB = BC = sqrt(29).

Step by step solution

01

Plot the points and form a triangle

Plot the given points A(1,-3), B(3,2), and C(-2,4) on a coordinate plane. Then, connect the points to form a triangle ABC.
02

Calculate the distances between points

Calculation of the distances between each pair of points A, B, and C using the distance formula. The distance between two points (x1, y1) and (x2, y2) can be calculated as: d = sqrt((x2 - x1)^2 + (y2 - y1)^2) - Subtract the coordinates of A and B: Distance AB = sqrt((3 - 1)^2 + (2 + 3)^2) = sqrt(2^2 + 5^2) = sqrt(29) - Subtract the coordinates of A and C: Distance AC = sqrt((-2 - 1)^2 + (4 + 3)^2) = sqrt(3^2 + 7^2) = sqrt(58) - Subtract the coordinates of B and C: Distance BC = sqrt((-2 - 3)^2 + (4 - 2)^2) = sqrt(5^2 + 2^2) = sqrt(29)
03

Check for equal distances

Now, we have the lengths of the three sides: AB = sqrt(29), AC = sqrt(58), and BC = sqrt(29). We see that AB = BC. Since two sides of the triangle have the same length, we have an isosceles triangle. As a result, the points A(1, -3), B(3, 2), and C(-2, 4) form an isosceles triangle.

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