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Squares and Square Roots

Squares and Square Roots

Word Problems

The Pythagorean Theorem is great for solving word problems. It's also good for sparking some scintillating dinner conversation. At least, we think so; no word yet from that dude who ran screaming from the room. When a word problem describes a situation that can be visualized using a triangle, it's likely that the Pythagorean Theorem is lurking somewhere nearby.

Even Darth Vader, a classic lurker, knows this.

Sample Problem

Juan and Leonor met for lunch. At 1 p.m., they parted ways. Maybe forever, considering how they left things. Juan drove due south at 30 mph and Leonor drove due east at 60 mph. Apparently, she was more upset than he was. At 1:30 p.m., how far away are Juan and Leonor?

This needs a picture, and there's definitely a triangle in here somewhere. Juan drove south, or straight down, and Leonor drove east, or straight out to the right:

The question is asking us to find c, and we've been given enough information to find a and b. The two have been traveling for half an hour...hopefully long enough for them to cool down and regret throwing all that furniture.

Leonor has traveled


and Juan has traveled


After half an hour, the triangle looks like this:

We're supposed to find the distance between Leonor and Juan, so we need to find c. We can do that with the Pythagorean Theorem:

(15)2 + (30)2 = c2,


1,125 = c2.

Taking the square root,

Negative distance doesn't make sense, although they were practically a negative distance away from each other a half hour ago when they were at each other's throats. Therefore, we take only the positive square root and conclude that, at 1:30 p.m., Juan and Leonor are miles apart (about 33.5 miles).

For the problem above, 33.5 miles would probably be an acceptable answer. Who cares if Juan and Leonor are closer to 33.541 miles apart? However, we gave the answer because that's an exact answer. We may not know all the decimal places of the number , but we know that's exactly how far apart they are at 1:30 p.m.

If a problem says to give an exact answer and your answer has radicals in it, leave the radicals there. The exact answer to the question "what is the square root of 2?" is , not 1.414. It may not surprise you to learn that math is something of an exact science, so those exact answers are important.

Note: Since the writing of this example, we at Shmoop are happy to report that Juan and Leonor have gotten together, talked, and worked things out. Not a single folding chair was hurled in the process.

Sample Problem

Find the exact area of a right triangle with hypotenuse 20 and one leg of length 7.

We'd better draw a picture, partly because we look for any excuse to stretch our artistic muscles. Use b for the unknown leg, and make that the base of the triangle:

We know the area of a right triangle is .

In other words, the area of a right triangle is one-half the product of the lengths of its legs. We know the length of one leg, and if we had the length of the other leg we'd be all set. Thankfully, we can use the Pythagorean Theorem to find the length of that missing piece. Otherwise, this triangle might need to be fitted for a peg leg.

We know that (b)2 + (7)2 = 202, so b2 = 351

Taking the positive square root (since a triangle can't have a negative base), we see that

This is great. If you write that down as your final answer, however, all your work will have been for nothing. We have not yet answered what the problem was asking us to find. Glancing back at the problem to remind ourselves what we were supposed to be looking for (oh, right, the area of the triangle), we see that we now have all the pieces we need. We know the height of the triangle is 7 and its base is , so its area is


That's a little nasty looking, but there is unfortunately no nicer way to write this. We're done. We wash our hands of you, ugly word problem.

Be careful: If you've read through the examples in this section and they made complete sense to you, give yourself a pat on the back—or a pat on the knee, if that's more easily accessible. You've gotten off to an excellent start.

Then, ask yourself this: would you know what to do if you were given similar problems and asked to solve them by yourself without looking at any webpages, books, or notes? Aye, there's the rub. Make sure you know how to approach the exercises, and any other problems available in your textbook or from your teacher, before deciding you are ready to stop studying. If the following exercises give you trouble, you can always review the above examples to figure out where you went wrong. They will always be there for you, like Ross, Rachel, Chandler, Monica, Joey and Phoebe. Okay, we'll throw Gunther in there, too.

Here's a hint for all of the exercises below: draw pictures. Bonus hint: the pictures should actually relate to the problem at hand. Art class is next period.

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