Question

$\huge\bold\red{Question}:-$
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If altitude of a right triangle is 7cm less than its base. If Hypotenuse is 13cm, find other two sides.
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Answer 1

AnswEr :

$\sf{Let\: Base\: of \: Right\: Triangle\: is \: x \; cm.}$

$\sf{Then, \: Altitude\: of \: Right\: Triangle\: is \: (x - 7) \; cm.}$

$\dag\:\: \small\bold{\underline{\sf{\red{Now,\:By\: using\: Pythagoras\: theorem}}}}$

$\longrightarrow\sf\: Base^2 \: + \; Altitude^2 \: = \: Hypotenuse^2$

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

$\:\:\:\:\:\: \footnotesize\bold{\underline{\sf{\pink{Using\; Identity\; (a \: -\: b)^2 \:=\:a^2\: + \: b^2 \:-\;2ab}}}}$

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

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

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

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

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

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

$\longrightarrow\sf\: (x - 12) = 0$

$\longrightarrow\bold{x \: = \: 12}$

$\longrightarrow\sf\: (x + 5) = 0$

$\longrightarrow\bold{x \: = \: -5}$

$\rule{150}2$

Side Can't be Negative.

Therefore, ( x = - 5) Neglected.

Altitude of Triangle = (x - 7) cm

$\longrightarrow\sf\: 12 - 7$

$\longrightarrow\large{\underline{\boxed{\sf{\red{x \: = \: 5 \: cm}}}}}$

Hence, Base of Triangle is 12 cm & Altitude is 5cm.

$\rule{150}2$

Answer 2

☯️ Given :-

• The altitude of a right triangle is 7 cm is less than its base
• Hypotenuse is 13 cm

☯️R.T.P :-

To find the other two sides

☯️ Solution :-

let the base be ' x '

According to Question,

altitude = x - 7 cm

Hypotenuse = 13 cm

☸️By Pythagoras therom , we get

${h}^{2} = {s}^{2} + {s}^{2}$

So, by substituting , we get

$( {13)}^{2} = {x}^{2} + ( {x - 7)}^{2}$

$169 = {x}^{2} + {x}^{2} - 14x + 49$

$169 = 2 {x}^{2} - 14x + 49$

$2 {x}^{2} - 14x + 49 - 169 = 0$

$2 {x}^{2} - 14x - 120 = 0$

$2( {x}^{2} - 7x - 60) = 0$

${x}^{2} - 7x - 60 = 0$

${x}^{2} - 12x + 5x - 60 = 0$

$x(x - 12) + 5(x - 12) = 0$

$(x - 12)(x + 5) = 0$

$x = 12 \: or \: - 5$

Side never be negative .

So, x = 12 cm

• Base = 12 cm
• Altitude = 12 - 7 = 5 cm
• Hypotenuse = 13 cm

☯️ I hope it helps you....✌️✌️