Question

the altitude of a right triangle is 7cm less than it's base. if the hypotenuse is 13cm, find the other two sides


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Answer 1

Answer:

answer is 5 cm of the altitude

Answer 2

Given:-

• The height of the triangle is 7cm less than its base.
• The Hypotenuse = 13cm

To Find:-

• The quadratic equation to find the base of the triangle

Solution:-

• Let the base of the triangle be x.In this question, we given that the height is 7cm less than the base.Then :-
• Height of the triangle will be = (x - 7)

$\small\underline{\pmb{\sf By \: applying\: Pythagoras \: Theorem:-}}$

$\sf \: \: \: \: \: \: \: \pink{Base^2 + Height^2 = Hypotenuse^2}$

$\sf \: \: \: \: \: \: \: : \implies x^2 + (x - 7)^2 = 13^2$

$\sf \: \: \: \: \: \: \: : \implies x^2 + (x)^2 -2(x)(7) + (7)^2 = 13^2$

$\sf\: \: \: \: \: \: \: : \implies x^2 + x^2 - 14x + 49 = 169$

$\sf \: \: \: \: \: \: \: : \implies 2x^2 - 14x + 49 - 169 = 0$

$\sf\: \: \: \: \: \: \: : \implies 2x^2 - 14x - 120 = 0$

$\sf \: \: \: \: \: \: \: : \implies 2(x^2 - 7x - 60) = 0$

$\sf \: \: \: \: \: \: \: : \implies x^2 - 7x - 60 = 0$

$\sf \: \: \: \: \: \: \: : \implies x^2 - 7x - 60 = 0$

$\sf \: \: \: \: \: \: \: : \implies x^2 -12x +5x - 60 = 0$

$\sf\: \: \: \: \: \: \: : \implies x(x-12)+5(x-12)$

$\sf \: \: \: \: \: \: \: : \implies (x-12)(x+5) = 0$

$\sf \: \: \: \: \: \: \: \pink{ : \implies x = 12,-5 }$

• (As side can't be negative.)

Hence,

• Base = (x) = 12 cm
• Height = (x - 7) = 12 - 7 = 5cm