Question

Two isosceles triangles have equal angles and their areas are in the ratio 16: 25. The ratio of corresponding heights is:​

Answer 1

Answer:-

Let the two triangles be ∆ABC & ∆DEF

IN ∆ABC

AB = AC [Given]

$\implies{\mathsf{ \frac{AB }{AC} = 1 }}⟹ 1$

Similiary in traingle DEF,

$\implies{\mathsf{\frac{DE}{DF}=1}}⟹ 2$

By equating both the equations (1) & (2)

$\implies{\mathsf{ \frac{ AB }{AC } = \frac{ DE }{DF }}}$

Now, This gives us

by

In ∆ABC & ∆DEF we get ∠A = ∠D

By using S.A.S (Side - angle - side) Congruency/similarity both triangles will

congruent.

$\boxed{\bold{\mathsf{S.A.S \: similarity}}}$

⟹ Any two triangles will be similar, if any one pair of

corresponding sides are proportional and the angles between them are equal.

∆ABC ~ ∆DEF

By using an basic property that is

★ Area's of any two similar triangle are in the ratio of the square of their

corresponding altitudes.

As AX & DY are the altitudes of the ∆ABC & ∆DEF respectively

$\implies{\mathsf{ \frac{ ar(ABC)}{ar(DEF } = \frac{ {AX}^{2} }{{DY}^{2} } }}$

$\implies{\mathsf{ \frac{ 16}{ 25 } = \frac{ {AX}^{2} }{{DY}^{2} } }}$

$\implies{\mathsf{\frac{ {4}^{2} }{ {5}^{2}} =\frac{ {AX}^{2} }{{DY}^{2} } }}$

$\huge{\boxed{\implies{\mathsf{\frac{ {AX} }{{DY} } = \frac{4 }{ 5 } }}}}$

So , The ratio of their corresponding heights is 4 : 5