Question

146. The perimeter of a right angled triangle is 24 cm.
If the hypotenuse is 10 cm, then area of this
triangle will be
(1) 20 cm2
(2) 22 cm2
(3) 24 cm
(4) 26 cm2​

Answer 1

Answer:

It is given that

Perimeter of a right angled triangle= 24 cm

its Hypotenuse= 10 cm

Now, in ∆ABC(See enclosed figure), applying Pythagoras theorem,

(Hypotenuse)²= (Base)²+(Perpendicular)²

(AC)²= (AB)²+(BC)²

(10)²= x²+y²

100= x²+y²--------(1)

We know that perimeter of a figure is the sum of all its sides,

Now,

Perimeter of ∆ABC= AB+BC+AC= x+y+10

But it is given that Perimeter= 24 cm,

So, x+y+10= 24

x+y= 24-10

x+y= 14-------(2)

We have an identity

(x+y)²= x²+ y²+ 2xy------(3)

Substituting (1) and (2) in (3),

(14)²= 100+2xy

196= 100+2xy

96= 2xy

96÷2= xy

48= xy

48/x= y

Putting y=48/x in equation (2),

x+48/x= 14

x²+48/x=14

x²+48= 14x

x²-14x+48=0

Now, factorising using middle term splitting

x²-6x-8x+48=0

x(x-6)-8(x-6)=0

(x-8)(x-6)=0

Therefore, x= 8 or x= 6

Substituting x=8 in (2),

8+y=14

y=14-8

y= 6 cm

Again putting x=6 in (2),

6+y=14

y=14-6

y= 8 cm

So, the other sides of the triangle are of lengths 6 cm and 8 cm.

We know that

Area of a triangle = 1/2×Base×height

= 1/2×6×8

= 6×4

= 24 cm²----Option 3

Answer=> 24 cm²