Hello,
To solve this problem we want to use the Pythagorean Theorem.
The pythagorean theorem states that for a 90° triangle,
[tex] a^{2} + b^{2} = c^{2} [/tex]
where a and b represent the two legs of the triangle, and c represents the hypotenuse.
Let a = the longer leg and b = the shorter leg.
If the longer leg of the triangle is 1 foot longer than the shorter leg, then
a = b +1.
If the hypotenuse is 9 feet longer than the shorter leg, then c = b + 9.
Using the equations we created, we can plug them into the Pythagorean Theorem to solve for a, b, and c.
Doing this, we have:
[tex] a^{2} + b^{2} = c^{2} [/tex]
[tex] (b+1)^{2} + b = (b+9)^{2} [/tex]
Expanding this, we get [tex] b^{2} + 2b + 1 + b^{2} = b^{2} + 18b + 81
2b^{2} + 2b + 1 = b^{2} + 18b + 81
b^{2} + 1 = 16b + 81
b^{2} = 16b + 80
b^{2} - 16b - 80 = 0
[/tex]
Solving for b, we get b = 20, and b = -4.
The length of the side of a triangle cannot be negative, so we know that b = 20.
However, we should check this with the original question to make sure it checks out.
a = b + 1
a = 20 + 1 = 21
c = b + 9
c = 20 + 9 = 29
So, we have a = 21, b = 20, and c = 29. (Also, 20-21-29 is a well known Pythagorean triple)
Using the Pythagorean Theorem, we have:
[tex] 21^{2} + 20^{2} = 29^{2} [/tex]
441 + 400 = 841
841 = 841, checks out.
So, the shorter leg is 20 feet, the longer leg is 21 feet, and the hypotenuse is 29 feet.
Hope this helps!
