Question

The altitude of a triangle is 6 cm greater than its base. If its area is 108 cm^2 , find its base.

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Base=12\:cm}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Altitude \: of \: triangle \: is \: 6 \: cm \: greater \: than \: base \\ \\ \tt: \implies Area \: of \: triangle = 108 { \: cm}^{2} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Base = ?$

• According to given question :

$\tt \circ \: Let \: base \: be \: x \\ \\ \tt \circ \: altitude = x + 6 \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Area \: of \: triangle = \frac{1}{2} \times Base \times Altitude \\ \\ \tt: \implies 108 = \frac{1}{2 } \times x \times (x + 6) \\ \\ \tt: \implies 216 = {x}^{2} + 6x \\ \\ \tt: \implies {x}^{2} + 6x - 216 = 0 \\ \\ \tt: \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \tt: \implies x = \frac{ - 6 \pm \sqrt{6^{2} - 4 \times 1 \times ( - 216)} }{2 \times 1} \\ \\ \tt: \implies x = \frac{ - 6 \pm \sqrt{36 + 864} }{2} \\ \\ \tt: \implies x = \frac{ - 6 \pm \sqrt{900} }{2} \\ \\ \tt: \implies x = \frac{ - 6 \pm30}{2} \\ \\ \green{\tt: \implies x = 12 \: cm} \\ \\ \green{\tt \therefore Base \: of \: triangle \: is \: 12 \: cm}$

Answer 2

Answer:

The Base is 12 cm.

Step-by-step explanation:

Let the base of a triangle be x cm.

Then,

Height = (x+6) cm

Area = 108 cm²

Value of x = Area = (Base*Height)/2

= 108 = x(x+6)/2

= 108 = x²+6x/2

= 216 = x²+6x

= 0 = x²+6x-216

= 0 = x²-12x+18x-216

= 0 = x(x-12)+18(x-12)

= 0 = (x-12)(x-18)

So,

The Value of x is either 12cm or, 18 cm.

Case 1: Taking x as 12 cm,

Height = 18 cm

Area = (12*18)/2

= 108 cm²

Case 2: Taking x as 18 cm,

Height = 24 cm

Area = (18*24)/2

= 216 cm²

So, as the Case 1 is matching with the question,

Base is 12 cm.

Hope this helps.