Question

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Answer 1

Answer :

The altitude of the triangle is 12cm and the base is 20cm

Given :

• The altitude of a triangle is three-fifth of the length of the corresponding base .
• When the altitude is decreased by 4cm and the corresponding base is increased by 10cm , the area of the triangle remains the same.

To Find :

• The base and the altitude of the triangle.

Formula to be used :

• $\sf{Area \: \: of \: \: a \: \: triangle = \dfrac{1}{2}\times Base \times Altitude \: \: (or Height) }$

Solution :

Let us consider the altitude of the triangle be x cm and corresponding base be y cm

Therefore,

$\bf{The \: \: area \: \: of \: \: triangle = \dfrac{1}{2} } \times x \times y$

According to Question :

The altitude of a triangle is three-fifth of the length of the corresponding base .

$\sf{\implies x = \dfrac{3}{5}\times y ...........(1)}$

When the altitude is decreased by 4cm and the corresponding base is increased by 10cm , the area of the triangle remains the same.

$\sf \dfrac{1}{2} \times (x - 4)(y + 10) = \dfrac{1}{2} xy \\ \\ \implies \sf(x - 4)(y + 10) = xy \\ \\ \implies \sf xy + 10x - 4y - 40 = xy \\ \\ \sf \implies10x - 4y - 40 = 0.......(2)$

Using the value of (1) in (2) we have :

$\sf10\{ \dfrac{3}{5} \times y \} - 4y - 40 = 0 \\ \\ \sf \implies6y - 4y - 40 = 0 \\ \\ \sf \implies2y = 40 \\ \\ \bf \implies y = 20$

And Now using the value of y in (1) :

$\sf{\implies x = \dfrac{3}{5}\times 20}\\\\ \implies \sf{x = 3\times4}\\\\ \implies\bf{x= 12}$

Thus the altitude is 12cm and the base is 20cm