Question

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Answer 1

question

prove that in an isosceles triangle the attitude from the vertex bisects the base

answer

In the above figure, there is an isosceles triangle whose two sides AB andBC have the same length.

Now, make some constructions in the figure to carry on the required proof. We know that the altitude from the vertex is perpendicular to the base, so we have to construct an altitude from the vertex B on the base AC which intersects AC at the point O. After the construction, the figure is given as:

Now, take a look in the two formed triangle ABO and CBO.

AB=BC (Equal sides of Isosceles triangle)

∠BOC=∠BOA(Angle formed by altitude is90∘)

BO=BO(Common side of triangles)

It can be seen that the two sides and the corresponding angles are the same so, using the SAS rule of congruence, we can say that the triangles ABO and CBO are congruent. That is,

ΔAOB≅ΔCOB

Therefore, we can conclude that the AO is equal to CO, which clearly shows that the altitude from the vertex bisects the base.

This is the required result that we have to prove.

Note:

This proof also gives another conclusion that the altitude from the vertex on the base bisects the angle formed at a vertex.