Question

Please help it’s so important

Answer 1

Given:

In $\Delta ABC, \overline{CA}=\overline{CB}$.

$\overline{CD}$ is an altitude drawn from C to $\overline{AB}$.

To prove:

$\overline{CD}$ bisects $\overline{AB}$.

Proof:

In $\Delta ABC$, $\overline{CD}$ is an altitude drawn from C to $\overline{AB}$.

It means, $\Delta ACD\text{ and }\Delta BCD$ are right angle triangles.

In $\Delta ACD\text{ and }\Delta BCD$,

Hypotenuse : $\overline{CA}=\overline{CB}$       [Given]

Leg : $\overline{CD}=\overline{CD}$       [Common]

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then by HL postulate both triangles are congruent.

$\Delta ACD\cong \Delta BCD$        [HL postulate]

$\overline{AD}\cong \overline{BD}$       [CPCTC]

$\overline{AD}=\overline{BD}$

It means, point D is the midpoint of $\overline{AB}$.

So, $\overline{CD}$ bisects $\overline{AB}$.

Hence proved.