Squares and Square Roots
Introduction to :
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.
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.
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.
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?
A square has area x. How long is a diagonal of the square?
Gina needs to use a ladder to retrieve a frisbee from the roof of her house. The frisbee is 12 feet above the ground, and because of the bushes near her house the base of the ladder will need to be 5 feet from the edge of the house. How long of a ladder does Gina need?
Leonardo walked north for 5 miles, south for 10 miles, and then east for 8 miles. How far away from his starting point did he finish walking?
A right triangle has a hypotenuse of 16 inches. The legs of the right triangle are the same length as each other. How long are the legs?
A square has area 2x + 4 and side length 10. Find x.
Find the radius of the circle shown below. The hypotenuse of the triangle has length 9.
Dorothy wants to build a triangular garden next to her house to go with her circular pool and trapezoidal deck. Her octagonal patio furniture is still on back order. The farthest the garden will extend from her house is 40 feet. One edge of the garden will be along the side of her house, which is L feet long. If Dorothy puts fencing along the other two edges of the garden, the ones that aren't the side of her house, how many feet of fencing will she need? Your answer may have L in it.
A boat left port and sailed due west for 2 hours at 10 mph, then sailed north for 3 hours. It didn't want to sail in a straight line to its destination, because it was trying to avoid another boat it dated a few months back. Awkward. The boat ended up 25 miles away from port as the crow flies (along a straight line). While the boat was sailing north, how fast was it sailing?
To drive from her house to her office, Fran drove north at 60 mph for 10 minutes, then turned and drove east at 48 mph. She didn't want to drive in a straight line to her destination, because, um, that's not how roads work. Her office is 26 miles from her house as the crow flies. Once again, that crow is flying in a straight line. How long did it take Fran to drive from her house to her office?
Johnny and his older, taller brother are standing 3 feet apart from each other. Johnny has already explained that this is how much room he needs for his "personal space bubble," and his brother can respect that. Johnny was trying to balance a perfectly straight, 4-foot stick on its end on top of his head, but he lost control of it and it fell, the opposite end of it landing on top of his brother's head. If his brother is 6 feet tall, how tall is Johnny?