Question

In triangle ABC,M and N are the mid points of Ab and AC respectively. The altitude AP to BC intersect MN at O. prove that AO = OP

Answer 1

Consider the triangle ABC, M and N are the midpoints of the sides AB and AC respectively.

Let AP be the altitude from the vertex A to the side BC.

The line joining the midpoints of AB and AC meet the altitude AP at O.

Required to prove that AO = OP.

So, we have to prove that AO = ½AP

In triangle ABC.

M and N are the midpoints of the sides AB and AC.

So, we have

AM = ½AB; AN = ½ AC

According to the Midpoint Theorem,

MN = ½ BC

So, ΔABC ~ ΔAMN

⇒ Area(AMN) / Area(ABC) = MN2 / BC2

= ( BC/2 )2 / BC2

= 1/4

= AO2 / AP2

So,

AO/AP = 1/2

AO = ½ AP

Hence, AO = OP.

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Answer 2

Answer:

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