The hypotenuse of a right triangle is $3\sqrt{10}$ cm. If the smaller leg is tripled and the longer leg doubled, new hypotenuse will be $9\sqrt5$ cm. How long are the legs of the triangle?

Given:

The hypotenuse of a right triangle is $3\sqrt{10}$ cm.

When the smaller leg is tripled and the longer leg doubled, new hypotenuse will be $9\sqrt5$ cm. 

To do:

We have to find the length of the legs of the triangle.

Solution:

Let the length of the smaller leg be $x$ cm and the length of the longer leg be $y$ cm.

Triple the length of the smaller leg$=3x$ cm.

Double the length of the longer leg$=2y$ cm.

We know that,

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. (Pythagoras theorem)

Therefore,

$x^2+y^2=(3\sqrt{10})^2$

$x^2+y^2=9(10)$

$x^2+y^2=90$----(1)

Also,

$(3x)^2+(2y)^2=(9\sqrt5)^2$

$9x^2+4y^2=81(5)$

$9x^2+4y^2=405$

$9x^2+4(90-x^2)=405$   (From equation 1)

$9x^2+360-4x^2=405$

$5x^2=405-360$

$5x^2=45$

$x^2=9$

$x=\sqrt9$

$x=3$ or $x=-3$

Length cannot be negative. Therefore, the value of $x$ is $3$.

$x^2=(3)^2=9\ cm^2$

$9+y^2=90$

$y^2=90-9$

$y^2=81$

$y=\sqrt{81}$

$y=9$

The lengths of the legs of the triangle are $3$ cm and $9$ cm. 

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Updated on: 10-Oct-2022

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