Question

$\large \pink{\underbrace{\bf{ \red{Question}}}}$

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Two isosceles triangles have equal vertical angles and their areas in the ratio 25: 36. Find the ratio of their corresponding heights. __________________________ Options:-
A-4:5
B-5:6
C-6:7
D-5:7 __________________________ Expecting answers from:-
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Need answer with explanation​

Answer 1

$\textrm{Option B, \rm 5:6 }$

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$\Large\textrm{Explanation}$

We know,

$\boxed{\rm\sin\theta=\dfrac{\rm(Opposite)}{\rm(Hypotenuse)}}$

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$\rm\therefore\sin\theta\times (Hypotenuse)=(Opposite)$

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The formula for the area of a triangle is,

$\boxed{\rm\dfrac{1}{2}\times(Base)\times(Height)}$

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We know the height we found previously.

The formula in terms of two sides $\rm a,\ b$ and an angle is,

$\boxed{\rm \dfrac{1}{2}ab\sin\theta}$

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Now, as a pair of two triangles are similar by AA similarity,

$\rm a:b=ak:bk\ (k>0)$

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Hence, the areas of two triangles become,

$\rm\dfrac{1}{2}ab\sin\theta,\ \dfrac{1}{2}\cdot ak\cdot bk\sin\theta$

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The ratios of the given area are,

$\rm1^{2}:k^{2}=5^{2}:6^{2}$

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We need to find $\rm1:k$.

Previously we found that,

$\rm1:k=5:6$

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Since corresponding heights are equal in ratio, the answer is,

$\textrm{ \cdots\longrightarrow\boxed{\textrm{Option B, \rm 5:6 }} }$