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Lorico [155]
2 years ago
6

Problem page the longer leg of a right triangle is 1ft longer than the shorter leg. the hypotenuse is 9ft longer than the shorte

r leg. find the side lengths of the triangle.
Mathematics
1 answer:
LenKa [72]2 years ago
7 0
Hello,

To solve this problem we want to use the Pythagorean Theorem. 
The pythagorean theorem states that for a 90° triangle, 

a^{2} +  b^{2}  =  c^{2}

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:
a^{2} +  b^{2} =  c^{2}
(b+1)^{2}  + b =  (b+9)^{2}

Expanding this, we get 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



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:

21^{2} +  20^{2} =  29^{2}
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!

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