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.