Question

Two isosceles triangle have equal vertical angles and tgeir ratio of their areas is 36:25 find the ratio of corresponding height

Answer 1

Answer:

Step-by-step explanation:

The Verticle angle is the angle other than the equal sides

Since, the verticle angles of both isosceles triangles is the same,

Let us consider there are 2 isoseles triangles, ΔABC and ΔPQR with verticle angles A and P respectively

∠A=∠P (Given)    ------>( 1 )

In an isosceles triangle, base angles are equal

i.e. ∠B=∠C and ∠Q=∠R    ----> ( 2 )

By Angle Sum Property

∠A+∠B+∠C=180° and ∠P+∠Q+∠R=180°

∠B+∠C=180-∠A and ∠Q+∠R=180-∠P

From ( 1 ) and ( 2 )

2∠B=180-∠A and 2∠Q=180-∠A

2∠B=2∠Q

∠B=∠Q    -------> ( 3 )

From ( 1 ) and ( 3 )

ΔABC~ΔPQR

By AA similarity

So, Ratio of their area= Ratio of square of their respective sides

36 : 25=AB² : PQ²

or √36/√25 = AB/PQ

So,  AB : PQ = 6 : 5   -----> ( 4 )

Let the height be AD and PS

∠ADB=∠PSQ=90° ( by construction )

∠B=∠Q ( proven earlier )

ΔABD~ΔPQS by AA similarity

So, AB/PQ=AD/PS

From ( 4)

AD : PS = 6 : 5

So, the ratio of the corresponding height of the triangles is 6 : 5

(Remember that we had only considered ΔABC and ΔPQR)

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