How do I solve this?

Question

The area of a right triangle is 96 square meters. The length of one leg is 8 meters less than twice the other. Find the length of each leg.

Answer 1

Let the length of one leg be x m
Therefore,the length of the other leg is
2x-8 meter
Therefore
1/2*x(2x-8)=96
or,x(x-4)=96
or,x^2-4x-96=0
or,(x-12)(x+8)=0
rejecting the negative value of x,we get x=12
Therefore the length of one leg=12 m
the length of the second leg=12*2-8=16 m

Answer 2

Assume that x is the length of one legs, the other one is 2*x - 8 (8 meter less than twice of this leg). Now that you have two legs you can calculate the area of the triangle :

a = x * (2*x-8) /2 = x^2 - 4*x and this equals to 96, thus:

x^2 - 4*x = 96 => x^2 - 4*x +4 = 100 => (x-2)^2 = 100 =>
x-2 = 10 => x=12

since the other leg is 2*x - 8, the length of the other leg is 2*12-8 which is 16

Good luck!

Answer 3

let the length of one leg = x
length of second leg = 2x -8
area = 1/2(first leg)(second leg) (not the hypotenuse)
96 = 1/2(x)(2x - 8)
96 = x(x - 4)
x^2 -4x -96 =0
x^2 -12x +8x - 96 = 0
x(x - 12) + 8(x - 12) = 0
(x - 12)(x + 8) =0
x = 12(ignoring negative value)
so length of the second leg = 2(12) -8 =16
so hypotenuse = sqrt(16^2 + 12^2)
= sqrt(256 + 144)
=sqrt(400)
= 20
the lengths of the legs are 12, 16, 20 meters

Answer 4

area of a triangle is A = 1/2BH
B= base, H = height
let:
B = 2x-8
H = x
since we are talking about a right triangle, the height is one of the legs.
96 = 1/2 * (2x-8)(x) A = 1/2 (B)(H)
192 = 2x²-8x ... 2x²-8x -192 = 0
solving for x's factor out a 2
2(x²-4x-96) = 0
2(x + 8)(x - 12) = 0
so one length is 8, the other is 12
x = 0, -8, 12
obviously neither leg can be -8, or 0 so our x_leg is 12
and 2(12)-8 = 16

check answer
A = 1/2BH
96 = 1/2(12)(16) true

Answer 5

A = bh/2 = 96 or bh = 2(96)

b = 2h-8

2(96) = (2h-8)h

2(96) = 2h^2 - 8h

h^2 - 4 h - 96 = 0

(h+8)(h-12) = 0

h = - 8 or h = 12; you cannot have a distance <0 so h = 12, b = 24-8 = 16

Answer 6

area of triangle formula is 1/2 the base x the height

[ (2 L -8) L ] / 2 = 96
2L^2 - 8L - 192 = 0
L^2 - 4L - 96 = 0
(L-12)(L+8) = 0

L=12 or -8 and -8 isn't possible
Base of triangle =12 Height of triangle = 16
Hypotenuse = 20
1/2(12)(8) = 96

Answer 7

the two legs are sitting on top of the base
Let h be the vertical leg, b the base
then the slanted leg = √(h^2 + b^2) = 2h -8

Area = hb/2 = 96
hb = 192

from √(h^2 + b^2) = 2h -8 = 2(h - 4)
h^2 + b^2 = 4 (h^2 - 8h + 16)

h √ [(2h-8)^2 - h^2] = 192

solves for h

Answer 8

12 and 16

Answer 9

area = 96 = 1/2 [ x (2x-8) ]
x^2 -4x - 96 = 0
(x-12)(x+8) = 0
x = 12 . . . and . . . L = 16

Answer 10

the 1st = x
the 2nd = 2x - 8

96 = x(2x-8)/2
96 = 2x(x-4)/2
96 = x(x-4)
96 = x^2 - 4x
x^2 - 4x - 96 = 0
x^2 + 8x - 12x - 96 = 0
x(x + 8) - 12(x + 8) = 0
(x - 12)(x + 8) = 0
x = 12
or
x = -8 (we cannt use this answer because it's minus)

so
the 2nd = 2x - 8
= 2(12) - 8
= 24 - 8
= 16

12, 16