Question

In a triangle ABC , ∠ A = 90°, side AB = x cm , AC = (x+2) cm and area=24cm . Find its perimeter.

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Answer 1

Answer:

Perimeter of triangle is 24 cm.

Step-by-step explanation:

Given:

In ΔABC, ∠A = 90° Area of Δ= 24 sq.cm.

AC=(x+2) cm., AB= x cm.

Since ∠A is 90° So ,BC is hypotenuse

Area of triangle = (1/2) × base × height

                  24 = (1 /2) × AC × AB

                 24 × 2 =(x+2) × x

                 48  =$x^{2}$ +2x

                 $x^{2}$+2x - 48=0

                 $x^{2}$ + 8x - 6x -48 =0       (by doing factorization)

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

                 (x - 6) (x +8)=0

Either x= 6 or x= - 8

Take x=6 as length is never negative .so ignore x= - 8

AB=x=6 cm. , AC= x + 2= 6 + 2=8 cm.

According to Pythagoras theorem,

$Hypotenuse^{2}$ =$perpendicular^{2}$ + $base^{2}$

          $BC^{2}$ =$AB^{2}$ +$AC^{2}$

              $BC^{2}$  = $6^{2}$ +$8^{2}$

                       =36 + 64

                       =100

                BC =$\sqrt{100}$

                       =10 cm.

Perimeter of Δ = AB + AC + BC

                         = 6 + 8 +10

                         =24 cm.

Answer 2

Answer:

The perimeter of the triangle is 24 cm.

Step-by-step explanation:

Given:

∠A = 90°

AB = x cm

AC = x + 2 cm

The area of the triangle = 24 cm²

To find out :

The perimeter of the triangle.

Solution:

To find the perimeter firstly we have to find the value of 'x'. Since one angle of the triangle, it is a right-angled triangle. The area of the right-angled triangle is given. The formula to find the area (A) is given as,

         $A=\dfrac{1}{2}bh$

Where 'b' is the base that is usually the bottom on which the triangle rests and 'h' is the height that is how tall the triangle is.

By substituting the values we get,

         $24=\dfrac{1}{2}\times x\times (x+2)$

       $24\times 2=( x^{2}+2x )\\\\48 = x^{2}+2x$

To solve this, add 1 on both sides. We get,

     48 + 1 = x² + 2x +1

Now we can write the RHS in the form of (a + b)². That is

       49 = (x + 1)²

   x + 1 = √49 = 7

         x = 7 - 1 = 6 cm

Hence the side AB = 6 cm. Therefore,

        AC = x + 2 = 6 + 2 = 8 cm

To find the perimeter we have to find the third side BC of the triangle which is the hypotenuse. According to the Pythagorean theorem, we know that hypotenuse² = adjascent² + opposite². Therefore,

       BC² = AC² + AB²

             = 8² + 6²

             = 64 + 36

       BC² = 100

        BC = √100 = 10 cm

Hence the perimeter (P) of the triangle is the sum of the three sides. Therefore, P = BC + AC + AB

                = 10 + 8 + 6

             P = 24 cm