Question

Find the area of a right-angled triangle whose hypotenuse is 41 cm and one side is 40 cm.
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Answer 1

Answer:

Sides are 41cm,40cm and 9cm

Area of a Right Angled Triangle = 1/2 BH

= 1/2 x 9 x 40 = 20 x 9 = 180 cm²

Answer 2

Answer :-

• The area of the right angled triangle is 180 cm².

To find :-

• The area of a right-angled triangle whose hypotenuse is 41 cm and one side is 40 cm.

Solution :-

• Before finding the area of the right-angled triangle, we have to find the length of it's other side. We will use the Pythagoras theorem for doing so as the length of the hypotenuse and one side of the triangle is given to us.

Pythagoras theorem states that :-

• The square of the hypotenuse is equal to the sum of the squares of the other two sides.

Here,

• Hypotenuse = 41 cm.
• One side = 40 cm.

Let :-

• The other side be "x" cm.

Then :-

• The square of the hypotenuse = 41².
• The square of one side = 40².
• The square of the other side = x².

Now, by Pythagoras theorem :-

$\implies\bf {x}^{2} + {40}^{2} = {41}^{2}$

$\bf\implies {x}^{2} + 1600 = 1681$

$\bf \implies {x}^{2} = 1681 - 1600$

$\bf \implies {x}^{2} = 81$

$\bf\implies x = \sqrt{81}$

$\implies \overline{ \boxed{\bf x = 9 \: cm}}$

Hence :-

• Length of the other side = 9 cm.

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• The sides of the right angled triangle are 41 cm, 40 cm and 9 cm.

It's a scalene triangle with all sides unequal, so we will use the Heron's formula for finding it's area, but before that, we have to find the semi-perimeter of the triangle!

We know that :-

$\underline{ \boxed{ \rm Semi\!-\!perimeter = \dfrac{a + b + c}{2}}}$

Where,

• a = First side.
• b = Second side.
• c = Third side.

Here,

• First side = 41 cm.
• Second side = 40 cm.
• Third side = 9 cm.

Therefore,

$\longrightarrow\bf Semi\!-\!perimeter = \dfrac{41 + 40 + 9}{2}$

$\bf \longrightarrow Semi\!-\!perimeter = \dfrac{90}{2}$

$\longrightarrow\overline{ \boxed{\bf Semi\!-\!perimeter = 45 \: cm}}$

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• Now let's find the area of the right angled triangle using Heron's formula!

Heron's formula states that :-

Area of a triangle is :-

$\bf \sqrt{s \: (s - a) \: (s - b) \: (s - c)}$

Where,

• s = Semi-perimeter.
• a = First side.
• b = Second side.
• c = Third side.

Here,

• Semi-perimeter = 45 cm.
• First side = 41 cm.
• Second side = 40 cm.
• Third side = 9 cm.

Therefore,

$\sf \hookrightarrow Area = \sqrt{45 \: (45 \!-\! 41) \: (45 \!-\! 40) \: (45 \!- \!9)}$

$\hookrightarrow\sf Area = \sqrt{45 \: (4) \: (5) \: (36)}$

$\hookrightarrow\sf Area = \sqrt{45 \times 4 \times 5 \times 36}$

$\hookrightarrow\sf Area = \sqrt{32400}$

$\hookrightarrow\overline{ \boxed{\sf Area = 180 \: cm^2}}$

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Hence :-

• The area of the right angled triangle = 180 cm².