Squaring the Triangle

Mentor: Today we will be working with right triangles. Before we start can you tell me what the definition of a triangle is?

Student: It's a three sided figure.

Mentor: Right, now knowing that can you tell me what a right triangle is?

Student: Well, a right angle is an angle that is 90 degrees, so wouldn't a right triangle be a triangle whose angles add up to 90 degrees?

Mentor: You were right about it being related to a right angle, but instead of all the angles adding up to 90 degrees, it means one of the angles in the triangle equals 90 degrees. It would be impossible for all the angles to add up to 90 degrees because in all triangles the angles must add up to 180 degrees.

Student: OK, so a right triangle is a triangle that includes a right angle.

Mentor: Good, now knowing what a right triangle is, what would be the sum of the other two angles?

Student: Well, if a triangle has 180 degrees total, and a right angle has 90 degrees, than the other two angles added together equal 90 degrees.

Mentor: Perfect. Now we know the basics of a right triangle. What we will be learning today is how to find length of the third side in a right triangle when the lengths of the other two sides are known.

Student: Why can't you just measure the length?

Mentor: Well, in some situations you can, but that process can be tedious and leaves room for human error. The mathematical process is much more accurate. Also, there are times when you cannot measure, such as when you don't have a ruler nearby, the triangle is not drawn to scale, or only numbers are given with no drawing to accompany it.

Student: OK, so if you can't rely on measuring, how would you do it?

Mentor: To find unknown sides in right triangles we use something called the Pythagorean Theorem. It is called this because it was created by a Greek philosopher and scientist named Pythagoras.

Student: What is the Pythagorean theorem and how do we use it?

Mentor: The Pythagorean theorem states that A ² +B ² =C ² where A represents one leg of the triangle, B represents the other leg, and C represents the hypotenuse, which is the side across from the right angle.

Student: OK, but how do we use this theorem since there are three variables involved?

Mentor: Well, remember at the beginning of the lesson I said that we would be finding the length of the third side of a triangle using the lengths of the other two sides.

Student: I remember now! So we can replace two of those variables with the sides we already know! But how do we know which lengths the variables represent?

Mentor: Well, if I label the sides of my triangle with A, B, and C where A and B are the legs of the triangle and C is the hypotenuse, how would this information be helpful when using the Pythagorean theorem?

Student: Well, it would make sense that A would be the length of side A and B would be the length of side B.

Mentor: Good, now lets find out why and how this theorem works and then we can practice using it a couple times. Lets go to the Squaring the Triangle .

Student: There's a triangle with a square attached to each side; is it a right triangle?

Mentor: Look over on the side where it says angle measurements and you tell me if it's a right triangle.

Student: It is! Angle C is 90 degrees.

Mentor: Right, in this activity the triangle is always a right triangle. We're going to use this activity to test out the Pythagorean theorem.

Student: But wait a second, the angles are labeled A, B, and C, I thought that the Pythagorean theorem works with the lengths of sides.

Mentor: You're right that the Pythagorean theorem works with the lengths of sides, but you need to remember that in different situations variables mean different things. It doesn't matter how the triangle is labeled, A, B, and C will always represent the sides.

Student: So in this case it would be (line AC) ² +(line BC) ² = (line AB) ² .

Mentor: Exactly.

Student: OK, so what are the squares attached to the triangle for? I thought the Pythagorean theorem only works for triangles.

Mentor: You're right that the Pythagorean theorem is only used for triangles, but if you look at the sides in the Pythagorean theorem you'll see that they are all squared. So, lets look at what really happens when you square a number. Get out a piece of graph paper and lets square 4.

Student: But that's easy, its 16.

Mentor: Right, but let's find out where the name squared came from in the first place.

Student: OK, so I've got a piece of graph paper what do I do?

Mentor: First take four boxes in a row on the graph paper and shade them in.

Student: Is it four because four is the number that we're trying to square?

Mentor: Right, if we were trying to square nine then you would shade in nine boxes in a row. Now, you know that four squared is just four times four, right?

Student: Yes, its always the number times itself.

Mentor: Right, and do you also remember that multiplication is really just adding numbers several times, so that 5*3 would in fact be 5+5+5?

Student: Yes, and so 2*5 would be 2+2+2+2+2.

Mentor: Or more simply it could be 5+5, but you've got the idea right. So, 4*4 would be what?

Student: Well, that would be 4+4+4+4 wouldn't it?

Mentor: Good, so we have the first four already written on the graph paper, so lets add 4 more in the row right below the first four. So now we've added 4+4. Now lets add another 4 in the row right below the second four, and we've got 4+4+4. Can you guess what we will do next?

Student: Add another 4 in the row below that!

Mentor: Good, and with all those shaded in, what have you got?

Student: It's a square!

Mentor: Right, that's where the term squared is from, because the result is in fact a square. To solidify this idea, try three squared.

Student: That's a square too!

Mentor: Good, so now can you guess why there are squares attached to the sides of the triangle in Squaring the Triangle?

Student: Well, it's called squaring the triangle so they've squared each side of the triangle.

Mentor: But why would the activity be called Squaring the Triangle, think about the Pythagorean theorem.

Student: Oh right! In the Pythagorean theorem A, B, and C are all squared, and A, B, and C, represent the sides of the triangle, so they squared the sides of the triangle.

Mentor: Exactly! This activity was designed to help people understand the Pythagorean theorem so it has drawn a diagram of what the theorem actually means. Now using those squares how can you check if the Pythagorean theorem is true?

Student: Well...if A ² +B ² =C ² shouldn't (the square attached to side A)+ (the square attached to side B)=(the square attached to side C)?

Mentor: Right, but which part of the square? The perimeter?

Student: Umm, I don't know.

Mentor: Well, lets go back to the square you drew on graph paper. You said before we started that 4 squared was 16 and you were right. So is the perimeter of the square you drew 16?

Student: Yea, it is!

Mentor: Just to make sure it's right, try it with the square you drew for three squared. So what's the perimeter of that square.

Student: It's 12...but three squared is 9.

Mentor: Right, this means that the fact that the perimeter of the first square being the same as the side squared was a fluke. The reason why it worked was because four squared is four times four, which is the area, and the length of the side is 4, and there's four sides so (the length of side*4), which is the perimeter. Now, that we know the perimeter is wrong what should we try next?

Student: How about the area?

Mentor: OK, try it with your four by four square.

Student: It's 16! But lets check it with the three by three square to make sure it's right.

Mentor: Good Idea.

Student: The area of the three by three square is nine! It works!

Mentor: OK, so now we've discovered that a number squared equals the area of its square. So what can we expect for the Pythagorean theorem now?

Student: It would be (the area of the square attached to side A)+(the area of the square attached to side B)=(the area of the square attached to side C).

Mentor: Perfect. So if you look on the left side of the pictures in the area labeled "area" you will see the area of the squares. What do you notice about them?

Student: Square AC+Square BC=Square AB.

Mentor: Since the area is labeled "area" you can tell that the numbers in that area are all areas. So does the Pythagorean theorem work?

Student: Yes!

Mentor: Be careful, we have already discovered that using only one piece of data can lead you wrong. Try changing the lengths of sides AC and BC using the sliders below the diagram, does it still work?

Student: Yes! I can't find any arrangement where it doesn't work.

Mentor: Good, now that we know that the Pythagorean theorem works and why it works we can now start solving problems using it. Do you remember what I said we use it for?

Student: To get the length of a side when you have the other two sides of a triangle.

Mentor: Almost right, this doesn't work for all types of triangles. What type of triangle does this work for?

Student: Oh right, it only works for right triangles.

Mentor: OK, so I'm going to give you the lengths of two sides and you tell me what the length of the third side is using the Pythagorean theorem. Getting a calculator would make this easier.

Student: OK, I've got a calculator; I'm ready.

Mentor: OK, I'll start with giving you the two legs and you have to find the hypotenuse. OK, one leg is 5 and the other is 7. What's the hypotenuse?

Student: OK, so 5 ² is 25 and 7 ² is 49, so 25+49=74. The hypotenuse is 74!

Mentor: Not quite, because remember that its A ² +B ² =C ² . You did the first half right, A ² +B ² , but when you got 74 you forgot that 74 only equals C ² , not C. So what would you do to finish solving this problem?

Student: Square 74?

Mentor: No, think about it algebraically, 74=C ² , so how would you solve to get C by itself?

Student: You take the square root of each side!

Mentor: Good, so finish solving the problem.

Student: OK, the square root of 74 is 8.6, so C=8.6.

Mentor: Great, lets try another one. How bout the legs are 3 and 4.

Student: OK, 3 ² is 9 and 4 ² is 16 so 9+16=25. Then the square root of 25 is 5!

Mentor: Nice work. Do you feel like you understand how to get the length of the hypotenuse when given the lengths of the two legs?

Student: Yes! What's next?

Mentor: Now we will try to find the length of the second leg when given the lengths of the first leg and the hypotenuse. So to do that we're going to have to do a little algebra. You can try it first, and then I will help if you run into any problems. So how about 15 and 7.

Student: Which one is the hypotenuse and which one is the leg?

Mentor: Well, the hypotenuse is always the longest side in a right triangle. This is because the longest side will be across from the largest angle, the second longest side across from the second largest angle, and shortest side across from the smallest angle. We know a triangle only has 180 degrees and that the hypotenuse has to be the longest side, because the largest that an angle can be in a right triangle other than the right angle is 89.999. So since the right angle is 90 degrees, and the largest any other angle can possibly be is 89.999 than the side across from the right angle has to be the longest side, also known as the hypotenuse. Knowing that can you tell me whether 15 or 7 is the length of the hypotenuse?

Student: It has to be 15.

Mentor: Right, because 15 is obviously longer than 7, now can you solve for the other leg?

Student: Well, that would be 7 ² is 49 and 15 ² is 225. So that would be 49+B ² =225. Do I take the square root of 225?

Mentor: No, that would be if we already had the two legs, this is where we use the algebra. How would we get the B alone on one side?

Student: Well you would subtract 49 from 225 so that would be 176. So B equals 176?

Mentor: No, because remember the hypotenuse has to be the longest side, and the hypotenuse is 15, so 176 is way too much. You need to finish the algebra to get B all the way alone.

Student: Oh right, now I take the square root of 176.

Mentor: Good, so what is the answer?

Student: It's 13.266.

Mentor: Great Job. Now you know what a right triangle is, what the Pythagorean Theorem is, why the Pythagorean Theorem works, and how to solve problems using the Pythagorean Theorem. Do you have any questions?

Student: You've told me how to find the third side if I have the first leg and the second leg, and how to find the third side if I have the hypotenuse and the first leg, but how do I do it if I have the hypotenuse and the second leg?

Mentor: It's the same as finding it when you have the hypotenuse and the first side. The variables A and B are interchangeable as long as you keep them constant.

Student: I get it! The legs are the same thing so you don't have to have different strategies for solving it.

Mentor: Right. Do you have any other questions?

Student: Nope!