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# The Hypotenuse of a Right Triangle is 3sqrt10. If the Smaller Leg is Tripled and the Longer Leg Doubled, New Hypotenuse Wll Be 9sqrt5. How Long Are the Legs of the Triangle? - Mathematics

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#### Question

The hypotenuse of a right triangle is 3sqrt10. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be 9sqrt5. How long are the legs of the triangle?

#### Solution

Let the length of smaller side of right triangle be x cm then larger side be y cm

Then, as we know that by Pythagoras theorem

x^2 + y^2 = (3sqrt10)^2

x2 + y2 = 90            .............. (1)

If the smaller side is triple and the larger side be doubled, the new hypotenuse is 9sqrt5 cm

Therefore,

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

9x2 + 4y2 = 405             ............. (2)

From equation (1) we get y2 = 90 - x2

Now putting the value of y2 in equation (2)

9x2 + 4(90 - x2) = 405

9x2 + 360 - 4x2 - 405 = 0

5x2 - 45 = 0

5(x2 - 9) = 0

x2 - 9 = 0

x2 = 9

x = sqrt9 = ± 3

But, the side of right triangle can never be negative

Therefore, when x = 3 then

y2 = 90 - x2

= 90 - (3)2

= 90 - 9

= 81

y=sqrt81

= ± 9

Hence, length of smaller side of right triangle be 3 cm then larger side be 9 cm

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The Hypotenuse of a Right Triangle is 3sqrt10. If the Smaller Leg is Tripled and the Longer Leg Doubled, New Hypotenuse Wll Be 9sqrt5. How Long Are the Legs of the Triangle? Concept: Solutions of Quadratic Equations by Factorization.