SOLUTION: The hypotenuse of a right triangle is 9cm less than two times the shortest leg. The longer leg is 36cm. Find the length of the shortest leg of the right triangle and find the hypo

Question 675205: The hypotenuse of a right triangle is 9cm less than two times the shortest leg.
The longer leg is 36cm. Find the length of the shortest leg of the right triangle and find the hypotenuse.
Please show all steps.
Thanks
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Let x = the length of the shortest leg. Then the length of the hypotenuse would be: 2x - 9.

We have expressions for all three sides of a right triangle. So they should fit in the Pythagorean equation:

Now we solve this for x. First we simplify:

Now we want one side to be zero. Subtracting both terms of the left side we get:

Now we factor (or use the Quadratic Formula). We can factor out the GCF of 3:

The second factor will factor, but not easily. (So I understand if you prefer to use the Quadratic Formula instead.)

From the Zero Product Property we know that one of these factors must be zero. The "3" will not be zero but the other two could:
x - 27 = 0 or x + 15 = 0
Solving there we get:
x = 27 or x = -15

Since x represents the shortest leg of our triangle, we will reject the negative answer. So the shortest leg is 27.

And we can use this value for x to find the hypotenuse:
2(27) - 9
54 - 9
45

So the shortest leg is 27 cm and the hypotenuse is 54 cm.

P.S. This problem can be solved very quickly if you are familiar with 3/4/5 right triangles. Not only will triangles with sides of 3, 4 and 5 be right triangles, so will any triangle that has multiples of 3, 4 and 5 for its sides.

Since the longer leg given to you, 36, is a multiple of 4 (the longer leg of 3/4/5 triangles), you might see if the other legs are the same multiples. 36 is 9 * 4. So let's pretend that the short leg is 9 * 3 = 27 and the hypotenuse is 9 * 5 = 45. All we have to do now is make sure that the hypotenuse is "9cm less than two times the shortest leg":
Is 2*27 - 9 = 45? Answer: Yes! So we could solve this very quickly this way. (NOTE: If the hypotenuse had not been "9cm less than two times the shortest leg" then we would have to solve this using algebra like the first, longer solution above.)
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