Question

QueStion:-

The altitude of a right triangle is 7cm less then its base. If the hypotenuse 13cm, Find the other two sides.​

Answer 1

GivEn:-

• The altitude of a right triangle is 7cm less then its base.

• Hypotenuse of right triangle is 13cm.

To find:-

• The other two sides of triangle.

SoluTion:-

★ Lets the base of right angle triangle be x cm.

Therefore, Height be (x - 7) cm.

★ Hypotenuse = 13cm

$\dag\;\bf{\underline{\pink{Using\; Pythagoras\; Theorem:-}}}$

$:\implies\sf (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2$

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

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

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

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

$\;\;\;\small\sf\dag\; \underline{Taking\;(2)\;common\;-}$

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

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

✠ Factorising The middle term -

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

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

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

$\;\;\;\small\sf\dag\; \underline{Both\;(x - 12)\;and\;(x + 5)\;equals\;to\;zero\;-}$

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

$:\implies\bf \blue{x = 12\; ; \;x = -5}$

Here, we gets two values 12 and -5.

But -5 can't be side of triangle becz, side can't be negative.

Therefore,

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

$\dag$ Hence, Base of triangle is 12cm and Height is 5cm.

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Additional Information:-

✩ A quadratic equation has three equal roots.

• Real and distinct.

• Real and coincident

• Imaginary

✩ If p(x) is a quadratic polynomial then p(x) = 0 is called Quadratic Polynomial.

★ General Formula: x² + bx + c = 0

✩ A polynomial whose degree will be 2 is considered as Quadratic Polynomial.

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Answer 2

GIVEN :

• The altitude of a right triangle is 7 cm less then its base.The hypotenuse is 13 cm.

TO FIND :

• The other two sides = ?

SOLUTION :

Let the x be the base of triangle.Then, the Altitude will be (x - 7)

★ By using Pythagoras theorem :

➨ x² + (x - 7)² = (13)²

➨ x² + x² + 49 - 14x = 169

➨ 2x² - 14x + 49 = 169

➨ 2x² - 14x + 49 - 169 = 0

➨ 2x² - 14x - 120 = 0

➨ x² - 7x - 60 = 0

➨ x² - 12x + 5x - 60 = 0

➨ x(x - 12) + 5(x - 12) = 0

➨ (x + 5) (x - 12) = 0

➨ x = -5 or x = 12 (The side of triangle never be negative)

So, x = 12

Hence, the base of triangle is 12 cm and the altitude of the triangle will be (x - 7) = 12 - 7 = 5 cm.