Question

Prove that a right triangle, the square on the hypotenuse is equal to the sum of the squares oh the square on the other two sides

Answer 1

Answer:

→ AC² = AB² + BC² .

Step-by-step explanation:

Given:

→ A right angled ∆ABC, right angled at B

To Prove:-

→ AC²=AB²+BC²

Construction:-

→  Draw perpendicular BD onto the side AC .

Proof:-

→ We know that if a perpendicular is drawn from the vertex of a right angle of a right angled triangle to the hypotenuse, than triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

We have

△ADB∼△ABC. (by AA similarity)

Therefore, AD/ AB=AB/AC

(In similar Triangles corresponding sides are proportional)

⇒ AB²=AD×AC……..(1)

Also, △BDC∼△ABC

Therefore, CD/BC=BC/AC

(in similar Triangles corresponding sides are proportional)

⇒ BC²=CD×AC……..(2)

Adding the equations (1) and (2) we get,

⇒ AB²+BC²=AD×AC+CD×AC

⇒ AB²+BC²=AC(AD+CD)

(∴ AD + CD = AC)

⇒ AB²+BC²=AC . AC

∴ AC²=AB²+BC² .

Hence, it is solved .

→ Note:- This theorem is also called as Pythagoras' theorem .

THANKS .

Answer 2

$\huge \bf{ \red{answer}}$

$\bf{question - }$
Prove that a right triangle, the square on the hypotenuse is equal to the sum of the squares oh the square on the other two sides

$\bf{step \: by \: step \: explanation}$

Given=

ABC is a right angled triangle
right angled at B.

To prove= AC^2=AB^2+BC^2

construction= BD perpendicular AC

Proof

$\bf \: in \triangle \: adb \: and \: \triangle \: abc$

/_A=/_A (common)

/_ADB=/_ABC. (both 90°)

$\bf{ \triangle \: adb \sim \triangle \: abc}$

by AA criteria.

$= > \: \frac{ad}{ab} = \frac{ab}{ac}$
$\sf{ = > \: ad \times ac = {ab}^{2}..........(1)}$
$\bf{in \triangle \: bdc \: and \triangle \: abc}$

/_C=/_C (common)

/_BDC=/_ABC (both 90°)

$\sf{ \triangle \: bdc \sim\: \triangle \: abc}$
$= > \: \frac{dc}{bc} = \frac{bc}{ac}$


$\sf \: = > dc \times ac = {bc}^{2} ..........(2)$


from equation(1) and (2) we get


AD*AC+DC*AC=AB^2+BC^2

(AD+DC)*AC=AB^2+BC^2

AC^2=AB^2+BC^2

HENCE, Proved

THANKS