Question

please solve this step by step with explanation.​

Answer 1

Given Question

In ∆ ABC, ∠C = 90°, CD ⊥ AB, CD =h, BC = a, CA = b, AB = c, Prove that

$\sf \: \dfrac{1}{ {h}^{2} } = \dfrac{1}{ {a}^{2} } + \dfrac{1}{ {b}^{2} }$

$\red{\large\underline{\sf{Solution-}}}$

Given that,

In ∆ABC, ∠C = 90° and BC = a, CA = b, AB = c

So, Area of ∆ ABC is given by

$\rm :\longmapsto\: Area_{∆ABC} = \dfrac{1}{2} \times ab - - - (1)$

Also,

In ∆ ABC, CD ⊥ AB

So, Area of ∆ABC is given by

$\rm :\longmapsto\: Area_{∆ABC} = \dfrac{1}{2} \times ch - - - (2)$

From equation (1) and (2), we get

$\rm :\longmapsto\:\dfrac{1}{2}ab = \dfrac{1}{2}ch$

$\rm :\longmapsto\:ab = ch$

$\rm\implies \:\boxed{\tt{ c = \frac{ab}{h}}} - - - (3)$

Again, In right ∆ ABC

Using Pythagoras Theorem, we have

$\rm :\longmapsto\: {a}^{2} + {b}^{2} = {c}^{2}$

On substituting the value of c, from equation (3), we get

$\rm :\longmapsto\: {a}^{2} + {b}^{2} = {\bigg[\dfrac{ab}{h} \bigg]}^{2}$

$\rm :\longmapsto\: {a}^{2} + {b}^{2} = \dfrac{ {a}^{2} {b}^{2} }{ {h}^{2} }$

can be rewritten as

$\rm :\longmapsto\:\dfrac{ {a}^{2} + {b}^{2} }{ {a}^{2} {b}^{2} } = \dfrac{1}{ {h}^{2} }$

$\rm :\longmapsto\:\dfrac{ {a}^{2}}{ {a}^{2} {b}^{2} } + \dfrac{ {b}^{2}}{ {a}^{2} {b}^{2} }= \dfrac{1}{ {h}^{2} }$

$\rm\implies \:\boxed{\tt{ \frac{1}{ {a}^{2} } + \frac{1}{ {b}^{2} } = \frac{1}{ {h}^{2} }}}$

Hence, Proved

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MORE TO KNOW

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

Answer 2

$\huge \color{red}\maltese \bold \color{yellow} \: answer$