Question 675205
Let x = the length of the shortest leg. Then the length of the hypotenuse would be: 2x - 9.<br>
We have expressions for all three sides of a right triangle. So they should fit in the Pythagorean equation:
{{{(x)^2 + (36)^2 = (2x-9)^2}}}<br>
Now we solve this for x. First we simplify:
{{{x^2 + 1296 = 4x^2-36x+81}}}
Now we want one side to be zero. Subtracting both terms of the left side we get:
{{{0 = 3x^2-36x-1215}}}
Now we factor (or use the Quadratic Formula). We can factor out the GCF of 3:
{{{0 = 3(x^2-12x-405)}}}
The second factor will factor, but not easily. (So I understand if you prefer to use the Quadratic Formula instead.)
{{{0 = 3(x-27)(x+15)}}}
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<br>
Since x represents the shortest leg of our triangle, we will reject the negative answer. So the shortest leg is 27.<br>
And we can use this value for x to find the hypotenuse:
2(27) - 9
54 - 9
45<br>
So the shortest leg is 27 cm and the hypotenuse is 54 cm.<br>
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.<br>
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.)