if the perimeter of a right triangle is 90cm and area is 270 sqcm, what is the length of hypotenuse?

Question

please give the working steps also.

Answer 1

Nishit V came close to giving the right answer, but unluckily he made a small calculation error. The actual working goes thus:

(90-z)^2 = z^2 + 4A

= 8100 + z^2 - 180z = z^2 + 1080
(Here Nishit V wrote 980 instead of 1080.)

=>180z = 8100 -1080 = 7020

=> z =7020/180

=> z = 39 cm

Answer 2

If the perimeter of a right triangle is 90cm and area is 270 sqcm, what is the length of hypotenuse
Let side be x and y so hypotenuse = z = (90 – x – y)
Area = (x × y) / 2 = 270
or x × y = 135
Now by Pythagoras theorem x^2 + y^2 = z^2
x^2 + y^2 = (90 – x – y)^2
on solving
x^2 + y^2 = (90) – (x + y)^2
x^2 + y^2 = 90^2 – 2 × 90 (x + y) + (x + y)^2
x^2 + y^2 = 8100 – 180 (x + y) + x^2 + 2x y + y^2 -------- as x y = 135, 2 x y = 270 putting this value in the equation
x^2 + y^2 = 8100 – 180 (x + y) + x^2 + 270 + y^2
Cancelling ( x^2 + y^2 ) on both the sides of equation
0 = 8100 – 180 (x + y) + 270
or 180 ( x + y) = 8370 -------------- x + y = 90 – z
180 ( 90 – z) = 8370
16200 – 180 z = 8370
or 180 z = 16200 – 8370 = 7830
or z = 7830 / 180 = 43.5
i.e. z, Hypotenuse = 43.5 cm

Answer 3

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Answer 4

lets a and b be the length of sides

The area is ab/2 = 270 so ab = 540 b=540/a (1)

the perimeter = a+b +(a+b) ^0.5 =90(2)

(2) can be rewritten a+b-90 = (a+b)^0.5 we will square this
a^2+b^2 +8100+2ab -180a-180 b = a+b replace b by (1)
a^2+(540)^2/a^2+8100+1080-180a-180*540/a= a+540/a
a^4+(540)^2+9180a^2-180*540a= a^3 +540a

I can not solve that expression but i have shown the way

Answer 5

Let sides be x,y and hypotenuse be z
x+y+z = 90 = P
1/2(xy) = 270 = A
and
z^2 = x^2 + y^2

Now (x+y)^2 = x^2 + y^2 + 2xy
or (P-z)^2 = z^2 + 4A
or (90-z)^2 = z^2 + 980
or 8100 + z^2 - 180z = z^2 + 980
180z = 8100 - 980 = 7120
=> z = 39.5556 cm

Answer 6

(90-z)^2 = z^2 + 4A

= 8100 + z^2 - 180z = z^2 + 1080
(Here Nishit V wrote 980 instead of 1080.)

=>180z = 8100 -1080 = 7020

=> z =7020/180

=> z = 39 cm

Answer 7

let assume two side of right trangle is a, b and hypotenus is c
perimeter is 90cm
then, a+b+c = 90 ----------(1)
area is 270 sq cm
area of right trangle = 1/2 * side1 * side 2
then, 1/2 *a*b = 270 ---------(2)
trangle abc is right angle trangle
then, c^2 = a^2 + b^2 ---------(3)

we solve the equation
though (2) and (3)
(a+b)^2 = (90-c)^2
c^2+1080 = 8100+ c^2 - 180c
c= 39 answer

Answer 8

let the 2 sides be l,b&hypotenuse be h
l+b+h=90
area =1/2*l*b
l*b=2*270=540 2*l*b=1080
l+b=90-h
(l+b)^2=(90-h)^2
l^2+b^2+2lb=8100+h^2-180h
l^2+b^2+1080=8100+h^2-180h
(l^2+b^2)=7020+h^2-180h
h^2=7020+h^2-180h
180h=7020
hypotenuse=39

Answer 9

using these two equations:
P=90=x+y+z ---------- (1)
A=270=(xy)/2 --------- (2)
x^2+y^2=z^2 ----------- (3)

(2) x=540/y
(1) z=90-(540/y)-y
(3) (540/y)^2 + y^2 = [90-(540/y)-y]^2
y=37.87

substitute y=37.87 to (1)
z (hypotenuse) = 37.87

Answer 10

the length of the hypoteus is 24300