Question

A triangle is isosceles if the altitude from the vertex angle to the base also passes through the midpoint of the base.

What are the two slopes from this figure that prove this? M is the midpoint of triangle ABC.

Answer 1

The point where the altitude intersects the base BC is (-4, 0), which is the midpoint of BC

Explanation:

To prove that triangle ABC is isosceles, we need to show that the altitude from vertex A to the base BC passes through the midpoint M of BC.

First, we can find the slope of the line segment BC using the coordinates of points B and C:

slope BC = (yC - yB) / (xC - xB) = (0 - 2) / (-2 - 2) = -1/2

Next, we can find the equation of the line perpendicular to BC that passes through point A. Since this line is perpendicular to BC and passes through point A, its slope will be the negative reciprocal of the slope of BC:

slope altitude = -1 / slope BC = -1 / (-1/2) = 2

Now we can use the slope of the altitude and the coordinates of point A to find the equation of the line that passes through A and has a slope of 2:

y - yA = 2(x - xA)

y - 4 = 2(x + 2)

y = 2x + 8

Finally, we can find the point where this line intersects the base BC by setting y = 0 and solving for x:

0 = 2x + 8

x = -4

So the point where the altitude intersects the base BC is (-4, 0), which is the midpoint of BC. Therefore, triangle ABC is isosceles.