Question

The perimeter og a right _ angled triangle is 48 centimeters, and its area is 96 square centimeters. length of hypotenuse is 20 cm.
What is the length of the perpendicular sides?​

Answer 1

AnSwer :

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Perimeter = 48 cm

Area = 96 cm²

If the Triangle is Right Angle. It would must Follow Pythagorean Triplets.

$\begin{gathered}\begin{tabular}{|c |c | c|}\cline{1-3}Perp/Base & Base/Perp & Hypotenuse \\\cline{1-3}3 & 4& 5 \\5 & 12 &13\\7 & 24&25 \\8 & 15&17\\\cline{1-3}\end{tabular}\end{gathered}$

So Either the Sides will any of these or Multiple of them. So by watching our Triangle Hypotenuse i.e. 20 cm we can find out that [ 3 , 4 , 5 ] pair will follow.

⠀

• Hypotenuse of Triangle is 4 times of the Pair (5 × 4) = 20 cm, Hence all sides will be 4 times too of the Pair.

⠀

• Perpendicular / Base = (3 × 4) = 12 cm

• Base / Perpendicular = (4 × 4) = 16 cm

⠀

Let's Check whether it's Correct or Not.

$\rule{160}{1.5}$

$\red{\underline{\bigstar\:\textsf{Perimeter of the Right Triangle :}} }</p><p>$

$\begin{gathered}:\implies\sf Perimeter=Perpendicular+Base+Hypotenuse\\\\\\:\implies\sf 48\:cm=12\:cm+16\:cm+20\:cm\\\\\\:\implies\sf 48\:cm=48\:cm\end{gathered} </p><p></p><p>$

⠀

$\red{\underline{\bigstar\:\textsf{Area of the Right Triangle :}} }</p><p>$

$\begin{gathered}: \gray{\implies\sf Area=\dfrac{1}{2} \times Base \times Perpendicular}\\\\\\: \gray{\implies\sf 96\:cm^2 = \dfrac{1}{2} \times16 \:cm \times 12 \:cm}\\\\\\: \gray{\implies\sf 96\:cm^2 = 8 \:cm \times 12 \:cm}\\\\\\: \gray{\implies\sf 96\:cm^2 = 96\:cm^2}\end{gathered} </p><p>$

Answer 2

AnSwer :

Correct Question :

The perimeter of a right angled triangle is 48 cm , and its area is 96 cm² . Length of its hypotenuse is 20 cm . What is the length of the perpendicular sides ?

Solution :

Let , In a right angled triangle ,

B ⇔ Base

P ⇔ Perpendicular

H ⇔ Hypotenuse

Perimeter of a right angled triangle is sum of all its sides ,

➳ Per. = B + P + H

Hypotenuse , H = 20 cm

➳ Per. = B + P + 20

But it is given Per. = 48 cm , so ,

➳ 48 = B + P + 20

➳ B + P = 28

➳ P = 28 - B ... (1)

Area of a right triangle is ,

Area of a right triangle is ,$\to \sf A=\dfrac{1}{2}\times b\times h$

where ,

where ,h denotes height = perpendicular (P)

where ,h denotes height = perpendicular (P)b denotes base = Base (B)

➠ A = ¹/₂ × B × P

Given , Area of the triangle is 96 cm²

➠ 96 = ¹/₂ × B × P

➠ BP = 192

Sub. (1) ,

➠ B ( 28 - B ) = 192

➠ 28B - B² = 192

➠ B² - 28B + 192 = 0

➠ B² - 12B - 16B + 192 = 0

➠ B ( B - 12 ) - 16 ( B - 12 ) = 0

➠ ( B - 12 ) ( B - 16 ) = 0

➠ B = 12 ; 16 cm

Sub. B value in (1) ,

➠ P = 28 - (12)  or 28 - (16)

➠ P = 16 or 12

➠ P = 16 ; 12 cm

So ,

So , Base of the triangle = 12 or 16 cm  $\green{\bigstar}$

Perpendicular of the triangle = 16 or 12 cm  $\pink{\bigstar}$

Length of the perpendicular sides are 12 and 16 cm respectively