Question

The longer leg of a right triangle is 4ft longer than the shorter leg. The hypotenuse is 8ft longer than the shorter leg. Find the side lengths of the triangle. Length of the shorter leg: _ft

Length of the longer leg: _ ft
Length of the hypotenuse: _ft

Answer 1

check the picture below


`\bf (s+8)^2=s^2+(s+4)^2 \\\\\\ s^2+16s+64=s^2+s^2+8s+16\implies 16s+64=s^2+8s+16 \\\\\\ 0=s^2-8s-48\implies 0=(s-12)(s+4)\implies \begin{cases} 0=s-12\\ \boxed{12=s}\\ ------\\ 0=s+4\\ -4=s \end{cases}`

it cannot be a negative value, since it's a side's length, thus is the positive one.

shorter leg is "s", longer one is "s+4" and hypotenuse is "s+8"

Answer 2

if the legs ofa right triangle are a and b and the hytponuse is c then
a²+b²=c²

lets say
a>b
so

longer leg is 4ft longer than shorter
a=4+b

hypotnuse is 8ft longer than shorter
c=8+b

rewrite teh equation in terms of b and solve
a²+b²=c²
subsitute
(b+4)²+b²=(b+8)²
b²+8b+16+b²=b²+16b+64
2b²+8b+16=b²+16b+64
minus (b²+16b+64) from both sides
b²-8b-48=0
what 2 numbers multiply to get -48 and add to get -8?
-12 and 4
(b-12)(b+4)=0
set equal to 0

b-12=0
b=12

b+4=0
b=-4
false, can't have negative length
b=12

so
a=4+b
a=4+12
a=16

c=8+b
c=8+12
c=2


shorter leg: 12ft
longer leg: 16ft
hypotonuse: 20ft