Question

If the point A(a, 2) is equidistant from the points B(8, −2) and C(2, −2), find the value of a.

Answer 1

The given points are A(a, 2), B(8, −2) and C(2, −2).
Also, it is given that AB = AC.
Therefore, (AB)² = (AC)²
$$\begin{gathered} \Rightarrow {\left(8-a\right)}^2+{\left(-2-2\right)}^2={\left(2-a\right)}^2+{\left(-2-2\right)}^2 \\ \Rightarrow {\left(8-a\right)}^2+{\left(-4\right)}^2={\left(2-a\right)}^2+{\left(-4\right)}^2 \\ \Rightarrow {\left(8-a\right)}^2+{\left(4\right)}^2={\left(2-a\right)}^2+{\left(-4\right)}^2 \\ \Rightarrow {\left(8-a\right)}^2+16={\left(2-a\right)}^2+16 \\ \Rightarrow {\left(8-a\right)}^2={\left(2-a\right)}^2 \\ \Rightarrow 64+a^2-16a=4+a^2-4a \\ \Rightarrow -12a=-60 \\ \Rightarrow a=\frac{-60}{-12}=5 \end{gathered}$$
Therefore, the value of a is 5.