# Triangle ABC has vertices A(3,1), B(5,7) and C(1, y). Find all y so that angle C is a right angle?

Sep 6, 2016

The two possible values of $y$ are $3$ and $5$.

#### Explanation:

For this problem, we need to consider AC be perpendicular to BC.

Since the lines are perpendicular, by the slope formula we have:

$\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = - \frac{{x}_{2} - {x}_{1}}{{y}_{2} - {y}_{1}}$

$\frac{y - 7}{1 - 5} = - \frac{1 - 3}{y - 1}$

$\left(y - 7\right) \left(y - 1\right) = 2 \left(- 4\right)$

${y}^{2} - 7 y - y + 7 = - 8$

${y}^{2} - 8 y + 15 = 0$

$\left(y - 3\right) \left(y - 5\right) = 0$

$y = 3 \mathmr{and} 5$

Hopefully this helps!