Question

1. a triangle has the vertices \\(a(-1, 5)\\) \\(b(7, -1)\\), \\(c(-2, 0)\\). find the equation of the altitude from point \\(c\\) to the side \\(ab\\).
2. a triangle has the vertices \\(a(8, -4)\\), \\(b(9, 5)\\) \\(c(3, -1)\\). find the equation of the altitude from \\(b\\)

Explanation:

Find the equation of the altitude from point C to side AB

Using the Gradient Formula knowledge point

$$ m_{AB} = \frac{-1 - 5}{7 - (-1)} = \frac{-6}{8} = -\frac{3}{4} $$

Using the Perpendicular Gradients knowledge point

$$ m_{\perp} = -\frac{1}{m_{AB}} = \frac{4}{3} $$

Using the point-slope form with point \(C(-2, 0)\):

$$
LATEXBLOCK0
$$

Find the equation of the altitude from point B to side AC

Using the Gradient Formula knowledge point

$$ m_{AC} = \frac{-1 - (-4)}{3 - 8} = \frac{3}{-5} = -\frac{3}{5} $$

Using the Perpendicular Gradients knowledge point

$$ m_{\perp} = -\frac{1}{m_{AC}} = \frac{5}{3} $$

Using the point-slope form with point \(B(9, 5)\):

$$
LATEXBLOCK1
$$

Answer:

Question 1

The equation of the altitude from point \(C\) to the side \(AB\) is:

$$y = \frac{4}{3}x + \frac{8}{3}$$

Question 2

The equation of the altitude from point \(B\) to the side \(AC\) is:

$$y = \frac{5}{3}x - 10$$