Check whether , and are the vertices of an isosceles triangle.

Coordinate Geometry | Exercise 7.1

Question 4

Check whether $(5, -2)$ , $(6, 4)$ and $(7, -2)$ are the vertices of an isosceles triangle.

Solution

Understand the Question

An isosceles triangle is a triangle that has at least two sides of equal length.
Let the given points be $A(5, -2)$ , $B(6, 4)$ , and $C(7, -2)$ .
To determine if $\Delta ABC$ is isosceles, we calculate the lengths of all three sides ( $AB$ , $BC$ , and $CA$ ) using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1 · Calculate the length of side AB

Let the points be $A(5, -2)$ , $B(6, 4)$ , and $C(7, -2)$ . Diagram 1

Using the distance formula for $A(5, -2)$ and $B(6, 4)$

\begin{aligned} AB &= \sqrt{(6 - 5)^2 + (4 - (-2))^2} \\ &= \sqrt{1^2 + (4 + 2)^2} \\ &= \sqrt{1^2 + 6^2} \\ &= \sqrt{1 + 36} \\ &= \sqrt{37} \text{ units} \end{aligned}

Step 2 · Calculate the length of side BC

Using the distance formula for $B(6, 4)$ and $C(7, -2)$

\begin{aligned} BC &= \sqrt{(7 - 6)^2 + (-2 - 4)^2} \\ &= \sqrt{1^2 + (-6)^2} \\ &= \sqrt{1 + 36} \\ &= \sqrt{37} \text{ units} \end{aligned}

Step 3 · Calculate the length of side CA

Using the distance formula for $C(7, -2)$ and $A(5, -2)$

\begin{aligned} CA &= \sqrt{(5 - 7)^2 + (-2 - (-2))^2} \\ &= \sqrt{(-2)^2 + (-2 + 2)^2} \\ &= \sqrt{(-2)^2 + 0^2} \\ &= \sqrt{4 + 0} \\ &= \sqrt{4} \\ &= 2 \text{ units} \end{aligned}

Step 4 · Compare side lengths

Comparing the lengths of the three sides $AB = \sqrt{37}, \quad BC = \sqrt{37}, \quad CA = 2$

Since $AB = BC = \sqrt{37} \ne CA$ , two sides are equal in length. Therefore, $\Delta ABC$ is an isosceles triangle.

Answer

Yes, $(5, -2)$ , $(6, 4)$ , and $(7, -2)$ are the vertices of an isosceles triangle.

Common Mistakes

Negative Sign Errors: Mishandling double negatives when subtracting coordinates, e.g., writing $4 - (-2)$ as $4 - 2 = 2$ instead of $4 + 2 = 6$ .
Squaring Negative Numbers: Incorrectly evaluating $(-6)^2$ as $-36$ instead of $+36$ .
Early Stopping: Calculating only two sides without verifying the third side (to confirm it is not equilateral, though equilateral triangles are technically also isosceles).