Geometry and Triangles

Question

The triangle $ABC$ is such that $AB = 12cm$ and $AC = 8cm$.

$X$ is the midpoint of the base $BC$.

If the area of the triangle is $72 cm^2$ what is the length of the perpendicular from $X$ to $AB$ and to $AC$?

Answer 1

We solve the problem of the length of the perpendicular from $X$ to $AB$, and leave the other one to you.

Join $A$ and $X$ by a line segment. Note that $\triangle XCA$ and $\triangle XBA$ have the same area. For with respect to the equal bases $XC$ and $XB$, these triangles have the same height.

Thus $\triangle XBA$ has area $36$. But $36$ is half the base $AB=12$ times the height, which is the length of the perpendicular from $X$ to $AB$.