Question

prove that in right angled triangle, square of hypotenuse is equal to the sum of squares of remaing two sides​

Answer 1

$\large\underline{\sf{Given- }}$

• A right-angled triangle ABC right-angled at B.

$\begin{gathered}\begin{gathered}\bf \: To\:prove - \begin{cases} &\sf{ \red{{AC}^{2} = {AB}^{2}+{BC}^{2}}}\end{cases}\end{gathered}\end{gathered}$

Construction :-

• Through B, Draw AD perpendicular to AC intersecting AC at D.

Proof :-

Now,

$\purple{\bf :\longmapsto\:In \: \triangle \: ADB \: and \: \triangle \: ABC}$

$\rm :\longmapsto\:\angle \: ADB = \angle \: ABC \: \: \: \: \red{ \{ \sf \: each \: 90 \degree \}}$

$\rm :\longmapsto\:\angle \: BAD = \angle \: BAC \: \: \: \: \red{ \{ \sf \: common \}}$

$\purple{\bf :\longmapsto \: \triangle \: ADB \: \sim\: \triangle \: ABC} \: \: \: \: \: \: \: \red{\sf\{AA \: similarity \}}$

$\rm :\implies\:\dfrac{AD}{AB} = \dfrac{AB}{AC}$

$\rm :\implies\:\boxed{ \red{ \bf \: {AB}^{2} = AD \times AC}} - - - (1)$

Now,

$\purple{\bf :\longmapsto\:In \: \triangle BDC \: and \: \triangle \: ABC}$

$\rm :\longmapsto\:\angle BDC \: = \: \angle \: ABC \: \: \: \: \: \red{ \{ \sf \: each \: 90 \degree \}}$

$\rm :\longmapsto\:\angle DCB \: = \: \angle \: BCA \: \: \: \: \: \red{ \{ \sf \: common \}}$

$\rm :\longmapsto\:\triangle BDC \: \sim \: \triangle \: ABC \: \: \: \: \: \: \: \: \red{ \{\sf \: AA \: similarity \}}$

$\rm :\implies\:\dfrac{DC}{BC} = \dfrac{BC}{AC}$

$\rm :\implies\:\boxed{ \red{ \bf \: {BC}^{2} = CD \times AC}} - - - (2)$

On adding equation (1) and equation (2), we get

$\rm :\longmapsto\:{BC}^{2}+{AB}^{2} = CD \times AC + AD \times AC$

$\rm :\longmapsto\:{BC}^{2}+{AB}^{2}=(CD + AD) \times AC$

$\rm :\longmapsto\:{BC}^{2}+{AB}^{2}=AC \times AC$

$\bf\implies \:\boxed{ \red{ \bf \:\:{BC}^{2}+{AB}^{2}= {AC}^{2}}}$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

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

Answer:

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