Question

Prove theorem8 . 1.

Short answer

If an altitude is drawn from the vertex opposite to the hypotenuse of a right triangle, then two triangles so formed are similar to the original triangle and also to each other.

Step-by-step solution

Step 1. Given information.

$\Delta ABC$ is a right angled triangle, $\angle B$ is of $90°$, and $BX$ is an altitude.

Step 2. Formula used.

The sum of internal angles of a triangle is always $180°$.

Step 3. Proof.

Consider the following figure:

Sum of internal angles of a triangle is $180°$.

So,

$\begin{array}{c}\angle p+\angle q+90°=180°\\ \angle p+\angle q=180°-90°\\ =90°\end{array}$

Angle $p$ and $q$ form a complementary pair, that is, one acute angle in right angle triangle will always be compliment of the other acute angle.

Now in $\Delta AXB$ and $\Delta BXC$,

$\begin{array}{l}\angle AXB~\angle BXC\left(Sincebotharerightangles\right)\\ \angle XBA~\angle XCB(Since\angle q=\angle XCB)\\ \angle XAB~\angle XBC(Since\angle p=\angle XAB)\end{array}$

Using $AAA$, the triangle$\Delta AXB~\Delta BXC$.

In $\Delta AXB$, $\Delta BXC$, and $\Delta ABC$all corresponding angles are congruent.

Therefore, by $AAA$criterion for similar triangles, $\Delta AXB~\Delta BXC~\Delta ABC$.