Name: Area of a Triangle (Coordinate Geometry)
Uploaded: 2019-07-19
Duration: 6 min 48 s
Description: Dozo, Mozo and Genius are looking at a globe and notice the Bermuda Triangle on it. Mozo is curious and wants to know the area of the triangle! Study how the...

Question

An equilateral triangle has two vertices at the points

(3, 4)

and

(- 2, 3)

, find the coordinates of the third vertex.

Accepted Answer

The distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by the formula
$d = \sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}$

In an equilateral triangle all the sides are of equal length.

Here, we are given that $A (3, 4)$ and $B (- 2, 3)$ are two vertices of an equilateral triangle.

Let $C (x, y)$ be the third vertex of the equilateral triangle.

$A B = \sqrt{(3 + 2)^{2} + (4 - 3)^{2}}$

$= \sqrt{(5)^{2} + (1)^{2}}$

$= \sqrt{25 + 1}$

$= \sqrt{26}$ units

$B C = \sqrt{(- 2 - x)^{2} + (3 - y)^{2}}$

$A C = \sqrt{(3 - x)^{2} + (4 - y)^{2}}$

We have $B C = A C$ .

$\sqrt{(- 2 - x)^{2} + (3 - y)^{2}} = \sqrt{(3 - x)^{2} + (4 - y)^{2}}$

Squaring on both sides we have,

$(- 2 - x)^{2} + (3 - y)^{2} = (3 - x)^{2} + (4 - y)^{2}$

$4 + x^{2} + 4 x + 9 + y^{2} - 6 y = 9 + x^{2} - 6 x + 16 + y^{2} - 8 y$

$10 x + 2 y = 12$

$5 x + y = 6$

Substituting $y = 6 - 5 x$ in $\sqrt{(- 2 - x)^{2} + (3 - y)^{2}}$ and equating it to $\sqrt{26}$ , we get

$\sqrt{26} = \sqrt{(- 2 - x)^{2} + (3 - 6 + 5 x)^{2}}$

Squaring on both sides,

$26 = (- 2 - x)^{2} + (- 3 + 5 x)^{2}$

$26 = 4 + x^{2} + 4 x + 9 + 25 x^{2} - 30 x$

$13 = 26 x^{2} - 26 x$

$1 = 2 x^{2} - 2 x$

$2 x^{2} - 2 x - 1 = 0$

$x = \frac{2 \pm \sqrt{4 - 4 (2) (- 1)}}{4}$

$= \frac{2 \pm \sqrt{12}}{4}$

$x = \frac{1 \pm \sqrt{3}}{2}$

Substituting $x = \frac{1 \pm \sqrt{3}}{2}$ in $y = 6 - 5 x$ , we get

$y = 6 - 5 (\frac{1 \pm \sqrt{3}}{2})$

$= \frac{12 - 5 \mp 5 \sqrt{3}}{2}$

$y = \frac{7 \mp 5 \sqrt{3}}{2}$

Thus, the two possible values of the third vertex are $(\frac{1 + \sqrt{3}}{2}, \frac{7 - 5 \sqrt{3}}{2})$ and $(\frac{1 - \sqrt{3}}{2}, \frac{7 + 5 \sqrt{3}}{2})$ .

An equilateral triangle has two vertices at the points $(3, 4)$ and $(- 2, 3)$ , find the coordinates of the third vertex.

Important Questions on Co-Ordinate Geometry

Learn and Prepare for any exam you want !

An equilateral triangle has two vertices at the points (3, 4) and (-2, 3), find the coordinates of the third vertex.

Important Questions on Co-Ordinate Geometry

Learn and Prepare for any exam you want !

An equilateral triangle has two vertices at the points $(3, 4)$ and $(- 2, 3)$ , find the coordinates of the third vertex.