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Please help it’s so important

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Sagot :

Given:

In [tex]\Delta ABC, \overline{CA}=\overline{CB}[/tex].

[tex]\overline{CD}[/tex] is an altitude drawn from C to [tex]\overline{AB}[/tex].

To prove:

[tex]\overline{CD}[/tex] bisects [tex]\overline{AB}[/tex].

Proof:

In [tex]\Delta ABC[/tex], [tex]\overline{CD}[/tex] is an altitude drawn from C to [tex]\overline{AB}[/tex].

It means, [tex]\Delta ACD\text{ and }\Delta BCD[/tex] are right angle triangles.

In [tex]\Delta ACD\text{ and }\Delta BCD[/tex],

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

Leg : [tex]\overline{CD}=\overline{CD}[/tex]       [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.

[tex]\Delta ACD\cong \Delta BCD[/tex]        [HL postulate]

[tex]\overline{AD}\cong \overline{BD}[/tex]       [CPCTC]

[tex]\overline{AD}=\overline{BD}[/tex]

It means, point D is the midpoint of [tex]\overline{AB}[/tex].

So, [tex]\overline{CD}[/tex] bisects [tex]\overline{AB}[/tex].

Hence proved.

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