Chapter 13: Q. 42 (page 527)

Discover and prove two things about the triangle with vertices K(-3, 4), M(3, 1), and J(-6, -2)

Short Answer

Expert verified

Triangle KMJ is an isosceles triangle.

Step by step solution

Step-1 – Given

Given that the triangle has vertices $K (- 3, 4), M (3, 1) and J (- 6, - 2)$ .

Step-2 – To determine

We have to discover and prove something about the triangle.

Step-3 – Calculation

We’ll use the distance formula to find the lengths of $K M, K J and M J$ .

The distance formula is given by: $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} [d = distance] So$

$\begin{array}{l} K M = \sqrt{{(3 - (- 3))}^{2} + {(1 - 4)}^{2}} [since, K (- 3, 4) and M (3, 1)] \\ K M = \sqrt{{(3 + 3)}^{2} + {(1 - 4)}^{2}} \\ K M = \sqrt{{(6)}^{2} + {(- 3)}^{2}} \\ K M = \sqrt{36 + 9} \\ K M = \sqrt{45} \\ K M = 3 \sqrt{5} \end{array}$

$\begin{array}{l} K J = \sqrt{{(- 6 - (- 3))}^{2} + {(- 2 - 4)}^{2}} [since, K (- 3, 4) and J (- 6, - 2)] \\ K J = \sqrt{{(- 6 + 3)}^{2} + {(- 2 - 4)}^{2}} \\ K J = \sqrt{{(- 3)}^{2} + {(- 6)}^{2}} \\ K J = \sqrt{9 + 36} \\ K J = \sqrt{45} \\ K J = 3 \sqrt{5} \end{array}$

$\begin{array}{l} M J = \sqrt{{(- 6 - 3)}^{2} + {(- 2 - 1)}^{2}} [since, M (3, 1) and J (- 6, - 2)] \\ M J = \sqrt{{(- 9)}^{2} + {(- 3)}^{2}} \\ M J = \sqrt{81 + 9} \\ M J = \sqrt{90} \\ M J = 3 \sqrt{10} \end{array}$

We see that the pairs of two sides are of equal lengths but the length of one side is different.

It means, triangle KMJ is an isosceles triangle.

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Discover and prove two things about the triangle with vertices K(-3, 4), M(3, 1), and J(-6, -2)

Short Answer

Step by step solution

Step-1 – Given

Step-2 – To determine

Step-3 – Calculation

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