Questions & Answers

Question

Answers

Answer

Verified

129k+ views

We have given triangle ABC and length of the side AB = 8 units. We need to find the lengths of the remaining sides ‘a’ and ‘b’ using the given condition a + b = 32.

Let us assume a + b = 32 be equation (1).

From the given figure we can see that the given triangle ABC is a right-angled triangle with right angle at B. We know that in a right-angled triangle, the side opposite to the right angle is known as hypotenuse and it is the largest side of the right-angled triangle.

According to the pythagoras theorem, we know that the sum of the squares of the remaining two sides of the right-angled triangle is equal to the square of the hypotenuse.

i.e., $ A{{B}^{2}}+B{{C}^{2}}=C{{A}^{2}} $ ---(2).

From the figure, we can see that the value of AB = 8 units, BC = a units and CA = b units.

We substitute these values in equation (2).

$ {{8}^{2}}+{{a}^{2}}={{b}^{2}} $ .

$ {{8}^{2}}={{b}^{2}}-{{a}^{2}} $ .

We know that $ \left( {{x}^{2}}-{{y}^{2}} \right)=\left( x-y \right)\times \left( x+y \right) $ .

$ 64=\left( b-a \right)\times \left( b+a \right) $ .

From equation (1),

$ 64=\left( b-a \right)\times 32 $ .

$ \left( b-a \right)=\dfrac{64}{32} $ .

$ \left( b-a \right)=2 $ .

$ b=2+a $ ---(3).

We substitute equation (3) in equation (1).

$ a+\left( 2+a \right)=32 $ .

2a + 2 = 32.

2a = 32 – 2.

2a = 30.

$ a=\dfrac{30}{2} $ .

a = 15.

We use the value of ‘a’ in equation (3) to get the value of ‘b’.

b = 2 + 15.

b = 17.

∴ The values of ‘a’ and ‘b’ are 15 and 17.