Question

6. ABC is a right-angled triangle. Angle ABC = 90°
AC = 25 cm and AB - 24 cm. Calculate the area of a
A.ABC​

Answer 1

Answer:

The area of the triangle is 84 cm².

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, △ABC is a right-angled triangle.

m∟ABC = 90°

AC = 25 cm - - - [ Given ]

AB = 24 cm

We have to find the area of the triangle.

In △ABC, m∟ABC = 90°

∴ ( AC )² = ( AB )² + ( BC )² - - [ Pythagoras theorem ]

⇒ ( 25 )² = ( 24 )² + ( BC )²

⇒ 625 = 576 + BC²

⇒ BC² = 625 - 576

⇒ BC² = 49

⇒ BC = √49 - - [ Taking square roots ]

⇒ BC = 7 cm

Now, we know that,

Area of right-angled triangle = ½ * Base * Height

⇒ A ( △ABC ) = ½ * BC * AB

⇒ A ( △ABC ) = ½ * 7 * 24

⇒ A ( △ABC ) = 7 * 12

⇒ A ( △ABC ) = 84 cm²

∴ The area of the triangle is 84 cm².

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Alternative Method:

In △ABC,

AB ( s₁ ) = 24 cm

BC ( s₂ ) = 7 cm

AC ( s₃ ) = 25 cm

Now, we know that,

Semi perimeter of triangle = ( Sum of sides ) / 2

⇒ s ( △ABC ) = ( s₁ + s₂ + s₃ ) / 2

⇒ s ( △ABC ) = ( 24 + 7 + 25 ) / 2

⇒ s ( △ABC ) = 56 ÷ 2

⇒ s ( △ABC ) = 28 cm

Now, we know that,

Area of triangle = √[ s ( s - s₁ ) ( s - s₂ ) ( s - s₃ ) ]

⇒ A ( △ABC ) = √[ 28 ( 28 - 24 ) ( 28 - 7 ) ( 28 - 25 ) ]

⇒ A ( △ABC ) = √( 28 * 4 * 21 * 3 )

⇒ A ( △ABC ) = √( 112 * 63 )

⇒ A ( △ABC ) = √[ ( 16 * 7 ) * ( 9 * 7 ) ]

⇒ A ( △ABC ) = √( 16 * 9 * 7 * 7 )

⇒ A ( △ABC ) = √( 4 * 4 * 3 * 3 * 7 * 7 )

⇒ A ( △ABC ) = 4 * 3 * 7

⇒ A ( △ABC ) = 12 * 7

⇒ A ( △ABC ) = 84 cm²

∴ The area of the triangle is 84 cm².

Answer 2

Given :-

• ∠ABC = 90°
• AC = 25 cm which is hypotenuse of triangle.
• AB = 24 cm which is base of triangle.

To find :-

• Area of △ ABC

At first we have to find side BC which is perpendicular.

So, length of BC is :- ( We have to find by Pythagoras theorem)

${ \boxed{\large{\mathtt{ \bigstar \: {(Hypotenuse)²= (Base)²+(Perpendicular)²}}}}}$

${\mathtt{\implies{(AC)² = (AB)² + (BC)²}}}$

${\mathtt{\implies{(25)²= (24)² + (BC)²}}}$

${\mathtt{\implies{625 = 576 + (BC)²}}}$

${\mathtt{\implies{625 - 576 = (BC)²}}}$

${\mathtt{\implies{(BC)² = 49}}}$

${\mathtt{\implies{BC = \sqrt{49} }}}$

${\boxed{\color{green}{\mathtt{\implies{BC = 7 cm}}}}}$

We know that

• Heron's Formula:-
• S = (a+b+c)/2 is the formula for finding the semiperimeter of triangle before finding the area of triangle.

• √[s(s – a)(s – b)(s – c)] is used to find the area when height is not given.

• Both are the formula of the Heron's formula for finding area of triangle.

So, semiperimeter of triangle is :-

${\sf{\implies{Semiperimeter = \frac{a + b + c}{2} }}}$

${\mathtt{\implies{S = \frac{AB + BC + CA}{2}}}}$

${\mathtt{\implies{S = \frac{24 + 7+ 25}{2}}}}$

${\mathtt{\implies{S = \frac{56}{2}}}}$

${ \boxed{ \color{green}{\mathtt{\implies{S = 28}}}}}$

So, Area of △ ABC is :-

${\mathtt{\implies{ \sqrt{[s(s – a)(s – b)(s – c)]} }}}$

${\mathtt{\implies{ \sqrt{[28(28 – 24)(28 – 7)(28 – 25)]} }}}$

${\mathtt{\implies{ \sqrt{[28(4)( 21)(3)]} }}}$

${\mathtt{\implies{ \sqrt{[28 \times 4 \times 21 \times 3]} }}}$

${\mathtt{\implies{ \sqrt{[7 \times 2 \times 2\times 2 \times 2 \times 7 \times 3 \times 3]} }}}$

${\mathtt{\implies{7×2×2×3}}}$

${\large{\boxed{\red{\mathtt{\implies{ 84 \: cm²}}}}}}$

So, Area of △ ABC is 84 cm².