Question

△abc is an equilateral triangle. what are the approximate coordinates of vertex a? a(x,y) b(2,1) c(6,1) a. (4, 2.5) b. (4, 3) c. (4, 4) d. (4, 4.5)

Explanation:

Step1: Find x - coordinate of A

The x - coordinate of the mid - point of BC gives the x - coordinate of A for an equilateral triangle. The formula for the mid - point of two points $(x_1,y_1)$ and $(x_2,y_2)$ is $x=\frac{x_1 + x_2}{2}$. Here, $x_1 = 2$ and $x_2=6$, so $x=\frac{2 + 6}{2}=4$.

Step2: Find length of BC

The length of BC using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Since $y_1=y_2 = 1$, $d=\sqrt{(6 - 2)^2+(1 - 1)^2}=\sqrt{4^2}=4$.

Step3: Find height of equilateral triangle

The height $h$ of an equilateral triangle with side length $s$ is $h=\frac{\sqrt{3}}{2}s$. Here $s = 4$, so $h=\frac{\sqrt{3}}{2}\times4 = 2\sqrt{3}\approx2\times1.732 = 3.464$.
The y - coordinate of A is $y=1 + h$. Since $h\approx3.464$, $y\approx1+3.464 = 4.464\approx4.5$.

Answer:

D. $(4,4.5)$