Question

Draw a median on the non congruent side of an isosceles triangle. Is it the altitude or an angle bisector too?
If true then enter 1 and if false then enter 0

Answer 1

Hint: An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. An angle bisector is a ray in the interior of an angle forming two congruent angles.

Complete step-by-step answer:
When two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there.

Consider triangle ABC, with AB = AC and AH perpendicular to BC.
Now, In triangle ABH and AHC,
AH=AH (Common)
\[\angle \]ABH=\[\angle \]ACH (Isosceles triangle property)
\[\angle AHB=\angle AHC={{90}^{\circ }}\,\text{(}AH\bot BC\text{)}\]
Thus, \[\vartriangle AHB\cong \vartriangle AHC\]
\[\angle \]BAH=\[\angle \]CAH (By cpct)
Thus, AH bisects \[\angle \]A
Moreover ,
\[\begin{align}
\Rightarrow & \angle AHC+\angle AHB={{180}^{\circ }} \\
\Rightarrow & \angle AHC=\angle AHB \\
\Rightarrow & \text{ 2}\angle AHC={{180}^{\circ }} \\
\Rightarrow & \angle AHC={{90}^{\circ }}\text{ } \\
\end{align}\]
It is an angle bisector as well as the altitude of the triangle. The answer is 1.
So, the correct answer is “1”.

Note: The median to unequal side of an isosceles triangle is always a perpendicular bisector of that side.
The reason is , the mentioned median divides the triangle into 2 congruent triangles. As lateral sides are equal being isosceles. Opposite side is bisected by the median. And one side is common. So the triangle will be congruent by SSS Congruence theorem So angles , where the median bisects, will be equal .But their sum = \[{{180}^{\circ }}\] . So each will be \[{{90}^{\circ }}\].
Hence the median to unequal side of an isosceles triangle is perpendicular as well as the bisector of the opposite side.