Question

an altitude of a triangle is five thirds of the length of its corresponding base. if the altitude is increased by 4 cm and the base be decrease by 2 cm,the area of triangle would remain the same. find the base and altitude of the triangle

Answer 1

Please check this answer

Answer 2

Given:-

The altitude of a triangle is five- thirds the length of its corresponding base. If the altitude is increased by 4 cm and the base decreased by 2 cm, the area of the triangle would remain the same.

To Find:-

The base and altitude of the triangle.

Solution:-

$\begin{gathered} \rm \: Let \: the \: lengt h \: of \: base \: of \: the \triangle \: b e \: x \: cm \\ \therefore \: \rm Altitude = \frac{5x}{3} \: cm \\ \rm \:Area \: of \triangle \: = \frac{1}{2} \bigg (base \times height \bigg) \\ \dashrightarrow \rm \: \frac{1}{2} \bigg(x \times \frac{5x}{3} \bigg) {cm}^{2} \end{gathered} </p><p>Letthelengthofbaseofthe△bexcm</p><p>∴Altitude= </p><p>3</p><p>5x</p><p> </p><p> cm</p><p>Areaof△= </p><p>2</p><p>1</p><p> </p><p> (base×height)</p><p>⇢ </p><p>2</p><p>1</p><p> </p><p> (x× </p><p>3</p><p>5x</p><p> </p><p> )cm </p><p>2</p><p> </p><p>$

When the altitude is increased by 4 cm and the base is decreased by 2 cm, we have

$\begin{gathered} \rm \:New \: base = (x - 2)cm \\ \rm New \: altitude = \bigg( \frac{5x}{3} + 4 \bigg)cm \\ \rm \:Area \: of \: new \triangle \\ \looparrowright \: \frac{1}{2} \rm\bigg \{ \bigg( \frac{5x}{3} + 4 \bigg) \times (x - 2) \bigg \} \\ \rm \looparrowright \: \frac{1}{2} \bigg \{ \frac{5 {x}^{2} }{3} - \frac{10x}{3} + 4x - 8 \bigg \} \\ \rm \looparrowright \: \frac{5 {x}^{2} }{6} - \frac{5x}{3} + 2x - 4 \end{gathered} </p><p>Newbase=(x−2)cm</p><p>Newaltitude=( </p><p>3</p><p>5x</p><p> </p><p> +4)cm</p><p>Areaofnew△</p><p>↬ </p><p>2</p><p>1</p><p> </p><p> {( </p><p>3</p><p>5x</p><p> </p><p> +4)×(x−2)}</p><p>↬ </p><p>2</p><p>1</p><p> </p><p> { </p><p>3</p><p>5x </p><p>2</p><p> </p><p> </p><p> − </p><p>3</p><p>10x</p><p> </p><p> +4x−8}</p><p>↬ </p><p>6</p><p>5x </p><p>2</p><p> </p><p> </p><p> − </p><p>3</p><p>5x</p><p> </p><p> +2x−4$

It is given that the area of the given triangle is same as the area of the new triangle. So,

$\begin{gathered} \therefore \: \rm \: \frac{5 {x}^{2} }{6} = \frac{5 {x}^{2} }{6} - \frac{5x}{3} + 2x - 4 \\ \rm \hookrightarrow \: \frac{5 {x}^{2} }{6} - \frac{5 {x}^{2} }{6} + \frac{5x}{3} - 2x = - 4 \\ \rm \hookrightarrow \: \frac{5x}{3} - 2x = - 4 \\ \rm \hookrightarrow \: 5x - 6x = - 12 \\ \rm \hookrightarrow \: - x = - 12 \\ \rm \hookrightarrow \: x = 12\end{gathered} </p><p>∴ </p><p>6</p><p>5x </p><p>2</p><p> </p><p> </p><p> = </p><p>6</p><p>5x </p><p>2</p><p> </p><p> </p><p> − </p><p>3</p><p>5x</p><p> </p><p> +2x−4</p><p>↪ </p><p>6</p><p>5x </p><p>2</p><p> </p><p> </p><p> − </p><p>6</p><p>5x </p><p>2</p><p> </p><p> </p><p> + </p><p>3</p><p></p><p>$

So, Base = 12cm

Altitude = 5x/3=5x12/3=20cm