Question

The hypotenuse of a right angle triangle is 25 cm and perimeter is 56 cm.find the smallest side using quadratic fromula

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Smallest\:side=7\:cm}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Hypotenuse = 25 \: cm \\ \\ \tt: \implies Perimeter \: of \: triangle = 56 \: cm \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt : \implies Smallest \: side = ?$

• According to given question :

$\circ \: \text{Let \: Base \: be \: x} \\ \\ \circ \: \text{Perpendicular \: be \: y} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Perimeter \: of \: triangle = a + b + c \\ \\ \tt: \implies 56 = 25 + x + y \\ \\ \tt: \implies x + y = 56 - 25 \\ \\ \tt: \implies x + y = 31 \\ \\ \tt: \implies y = 31 - x \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies {h}^{2} = {p}^{2} + {b}^{2} \\ \\ \tt: \implies {25}^{2} = {y}^{2} + {x}^{2} \\ \\ \tt: \implies 625 = {(31 - x)}^{2} + {x}^{2} \\ \\ \tt: \implies 625 = 961 + {x}^{2} - 62x + {x}^{2} \\ \\ \tt: \implies 2 {x}^{2} - 62x + 961 - 625 = 0 \\ \\ \tt: \implies {x}^{2} - 31x + 168 = 0 \\ \\ \tt: \implies {x}^{2} - 24x - 7x + 168 = 0 \\ \\ \tt: \implies x(x - 24) - 7(x - 24) = 0 \\ \\ \tt: \implies (x - 7)(x - 24) = 0 \\ \\ \green{\tt: \implies x = 7 \: and \: 24} \\ \\ \green{\circ\:\tt Base = 7 \: cm }\\ \\ \green{\circ\:\tt Perpendicular = 31 - 7 = 24 \: cm} \\ \\ \green{\circ\:\tt Base = 24 \: cm} \\ \\ \green{\circ\:\tt Perpendicular = 31 - 24 = 7 \: cm} \\ \\ \green{ \tt\therefore Smallest \: side \: length \: is \: 7 \: cm}$

Answer 2

$\huge \mathtt{ \fbox{Solution :)}}$

Given ,

• Hypotenuse of the triangle = 25 cm
• Perimeter of the triangle = 56 cm

Let ,

• Base and perpendicular of the triangle be x and y

We know that , perimeter of the triangle is given by

$\large \mathtt{ \fbox{Perimeter = A + B + C}}$

Thus ,

56 = 25 + x + y

x + y = 31

x = 31 - y

And the square of the longest side of triangle is

$\large \mathtt{ \fbox{ {(h)}^{2} = {(b)}^{2} + {(p)}^{2} }}$

Thus ,

$\sf \hookrightarrow {(25)}^{2} = {(x)}^{2} + {(31 - x)}^{2} \\ \\ \sf \hookrightarrow </p><p>625 = {(x)}^{2} + {(31)}^{2} + {(x)}^{2} - 2 × 31x \\ \\ \sf \hookrightarrow </p><p>625 = 961 +2 {(x)}^{2} - 62x \\ \\ \sf \hookrightarrow </p><p>2 {(x)}^{2} - 62x+ 336 = 0 \\ \\ \sf \hookrightarrow </p><p> {(x)}^{2} - 31x + 168 = 0 \\ \\ \sf \hookrightarrow </p><p>x(x - 24) - 7(x - 24) \\ \\ \sf \hookrightarrow </p><p>(x - 7)(x - 24) \\ \\ \sf \hookrightarrow </p><p>x = 7 \: \: cm \: or \: x = 24 \: \: cm$

Hence , The smallest side of the triangle is 7 cm or 24 cm

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