The length of one leg of a right triangle is proportional to the length of the other leg, with constant of proportionality 5. If the hypotenuse of the triangle has length 100, what are the exact lengths of the legs?
Once again, we're dealing with a triangle, so we'll draw one and fill in what we know. The hypotenuse has length 100, and if one leg has length x the other leg has length 5x, since that's the meaning of "proportional with constant of proportionality 5.' This triangle will be awfully wobbly when it walks.
The Pythagorean Theorem tells us that
(5x)2 + x2 = 1002,
which simplifies to
26x2 = 1002.
For this problem, it's easier to write 1002 than it is to write 10,000. Although now that we've gone ahead and written it out anyway in this explanation, it kind of defeats the purpose.
x2 = 1002/26,
The short leg of the triangle has length
and the longer leg has length
Since we were asked to give exact answers, we'll leave the answers like that. You wouldn't want to give those measurements to a carpenter who is cutting a triangle shape out of plywood, but it's all we need to answer this problem. However, it's fine to use the calculator to check that our answers are reasonable.
The shorter leg of the triangle has a length of approximately 19.6 and the longer leg has a length of approximately 98. Since our calculator informs us that, (19.6)2 + 982 = 9,988.16, which is fairly close to 1002, our answers seem reasonable. Now we know what information to give our carpenter friend.