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:

hope it will help you

Step-by-step explanation:

Given: Area =96 cm

2

and Perimieter =48cm

Let a,b,c be the sides of a right triangle, with c as hypotenuse.

Area=

2

1

ab=96

ab=192

b=

a

192

Perimeter=a+b+c=48

c=48−a−b

From Pythagoras Theorem,

c

2

=a

2

+b

2

(48−a−b)

2

=a

2

+b

2

2304+(a+b)

2

−96(a+b)=a

2

+b

2

2304+a

2

+b

2

+2ab−96(a+b)=a

2

+b

2

2304+2ab−96a−96b=0

Divide by 2,

1152+ab−48a−48b=0

1152+a(

a

192

)−48a−48(

a

192

)=0

1344−48a−

a

9216

=0

48a

2

−1344a+9216=0

a

2

−28a+192=0

a=16 and a=12

There are 2 possibilities as we have 2 perpendicular side

If, a=16, b=

a

192

=

16

192

=12

If, a=12, b=

a

192

=

12

192

=16

If, a=12 then b=16 or if b=12 then a=16

It doesn't make a difference while computing c

c

2

=a

2

+b

2

c

2

=144+256=400

c= Hypotenuse =20 cm

Answer 2

Step-by-step explanation:

Answer:

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\linethickness{0.23mm}\put(7.7,2.9){\large\sf{A}}\put(7.7,1){\large\sf{B}}\put(10.6,1){\large\sf{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\qbezier(10.5,1)(10,1.4)(8,2.9)\put(9.4,1.9){\sf{\large{20 cm}}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

⠀

Perimeter = 48 cm

Area = 96 cm²

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

⠀

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

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.

$\rule{180}{1.5}$

$\underline{\bigstar\:\textsf{Perimeter of the Right Triangle :}}$

$:\implies\sf Perimeter=Perpendicular+Base+Hypotenuse\\\\\\:\implies\sf 48\:cm=12\:cm+16\:cm+20\:cm\\\\\\:\implies\sf 48\:cm=48\:cm$

⠀

$\underline{\bigstar\:\textsf{Area of the Right Triangle :}}$

$:\implies\sf Area=\dfrac{1}{2} \times Base \times Perpendicular\\\\\\:\implies\sf 96\:cm^2 = \dfrac{1}{2} \times16 \:cm \times 12 \:cm\\\\\\:\implies\sf 96\:cm^2 = 8 \:cm \times 12 \:cm\\\\\\:\implies\sf 96\:cm^2 = 96\:cm^2$