Question

if hypotenous of right angled isosceles triangle is 4√2cm. find it's area.​

Answer 1

Given :-

If hypotenous of right angled isosceles triangle is 4√2cm.

To find :-

Area of isosceles triangle

Explanation :-

• Isosceles triangle

A triangle having two equal side and one unequal side.

• Equilateral triangle

A triangle having three equal side

• Scalene trianlge

All three side of this triangle are unequal.

Solution :-

• Hypotenuse of right angled isosceles triangle = 4√2 cm

According to Pythagoras theorem

(hypotenuse)² = (base)² + (perpendicular)²

• Let the each equal side of isoceles trianlge be x

→ (AC)² = (BC)² + (AB)²

→ (4√2)² = (x)² + (x)²

→ 32 = x² + x²

→ 32 = 2x²

→ x² = 32/2

→ x² = 16

→ x = √16

→ x = ± 4 cm

ɴᴏᴛᴇ

• Length never in negative

Hence,

• Measure of each equal sides of isosceles triangle = 4 cm

As we know that

→ Area of trianlge

→ ½ × base × height

→ ½ × AB × BC

→ ½ × 4 × 4

→ 2 × 4

→ 8 cm²

•°• Area of isoceles trianlge is 8cm²

Answer 2

$\underline{\underline{\sf{\clubsuit \:\:Question}}}$

• If hypotenuse of right angled isosceles triangle is 4√2 cm. Find it's area​

$\underline{\underline{\sf{\clubsuit \:\:Given}}}$

• Hypotenuse of right angled isosceles triangle is 4√2 cm

$\underline{\underline{\sf{\clubsuit \:\:To\:\:Find}}}$

• Area of right angled isosceles triangle

$\underline{\underline{\sf{\clubsuit \:\:Answer}}}$

• Area of riangle = 8 cm²

$\underline{\underline{\sf{\clubsuit \:\:Calculations}}}$

We know :

• An Isosceles Right angled Triangle is a right angled triangle that consists of two equal length sides

• In an isosceles right triangle, two sides and the two acute angles are congruent

Let us consider an isosceles right angled triangle ABC

In which :

• Hypotenuse = AC = 4√2 cm
• 1st side = AB
• 2nd side = BC

We already came to know that 1st side and 2nd side are equal in an isosceles right angled triangle

To calculate triangle's area, we need to know about Pythagoras theorem

Pythagoras theorem :

The Pythagoras theorem states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the sides of the right triangle

(Hypotenuse)² = (Side)² + (Side)²

(AC)² = (AB)² + (BC)²

(4√2 cm)² = (AB)² + (BC)²

4²(√2)² cm² = (AB)² + (BC)²

16 × 2 cm² = (AB)² + (BC)²

32 cm² = (AB)² + (BC)²

As two sides are equal,

We can consider 1st side and 2nd second sides as AB

32 cm² = (AB)² + (AB)²

32 cm² = (AB)² + (AB)²

32 cm² = 2(AB)²

Divide both sides by 2 :

32 cm²/2 = 2(AB)²/2

16 cm² = (AB)²

Take Square Root on both sides :

√(16 cm²) = √(AB)²

± 4 cm = AB

Since length can be never negative

Each equal side = 4 cm

So :

• AC = 4√2 cm

• AB = 4 cm

• BC = 4 cm

We also know :

$\boxed{\boxed{\bf{Area\:\:of\:\:Triangle\:\:=\:\:\dfrac{1}{2}}\:\:\times\:\:Base\:\:\times\:\:\ Height}}$

In our cases , we have :

• Base = BC  = 4 cm

• Height = AB  = 4 cm

$\bf{Area\:\:of\:\:Triangle\:\:=\:\:\dfrac{1}{2}}\:\:\times\:\:Base\:\:\times\:\:\ Height$

$\implies\sf{Area\:\:of\:\:Triangle\:\:=\:\:\dfrac{1}{2}\:\:\times\:\:4\:cm\:\:\times\:\:\ 4\:cm}$

$\implies\sf{Area\:\:of\:\:Triangle\:\:=\:\:\dfrac{1}{2}\:\:\times\:\:16\:cm^2}$

$\implies\sf{Area\:\:of\:\:Triangle\:\:=\:\:\dfrac{16}{2}\:cm^2}$

$\boxed{\implies\bf{Area\:\:of\:\:Triangle\:\:=\:\:8\:cm^2}}$