Question

In a right-angled triangle, prove that the hypotenuse is the longest side.​

Answer 1

Answer:

Let’s name triangleABC

In triangle ABC

Angle B=90

0

Angle B and angleA,

AC>BC ………..(1) (side opposite to greater angle is longer)

AngleB and angle c,

AC>AB ……..(2) (side opposite to greater angle is longer)

From equation 1st and 2nd ,

AC>BC,AB

Hence, hypotenuse is the longest side.

Answer 2

Step-by-step explanation:

Given :-

A right-angled triangle

Required To Prove :-

In a right-angled triangle, the hypotenuse is the longest side.

Proof :-

∆ ABC is a right angled triangle.

Right angle is at ∠B

∠B = 90°

The opposite side to ∠B = AC

Hypotenuse = AC

We know that

The sum of all angles in a triangle is 180°

=> ∠ A + ∠ B + ∠C = 180°

=> ∠ A + 90° + ∠C = 180°

=> ∠ A + ∠C = 180°-90°

=> ∠ A + ∠C = 90°

=> ∠A + ∠C = ∠B ---------(1)

So , ∠A and ∠C are acute angles which are less than ∠B each.

Now,

∠A < ∠B => BC < AC

Since Side opposite to greater angle is longer

Therefore, AC > BC -------------(2)

and

∠C < ∠B => AB > AC

Since Side opposite to greater angle is longer

Therefore, AC > AB ---------------(3)

From (2)&(3)

AC > AB and AC

AC is longer than both AB and AC

=> AC is the longer than remaining two sides.

=> AC is the longest side in ∆ABC.

=> Hypotenuse is the longest side in ∆ ABC

Hypotenuse is the longest side in the right angled triangle.

Hence, Proved.

Used formulae:-

• The sum of all angles in a triangle is 180°

• Side opposite to greater angle is longer