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## 8th grade (Illustrative Mathematics)

### Course: 8th grade (Illustrative Mathematics) > Unit 8

Lesson 10: Lesson 10: Applications of the Pythagorean Theorem# Pythagorean theorem word problem: carpet

CCSS.Math:

If we draw the diagonal on a rectangular carpet, then its length, width, and diagonal form a right triangle. Since we know the length and diagonal measurements, we can use the Pythagorean theorem to solve for the missing width of the carpet. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- what is meant by diagonal?(2 votes)
- Draw a rectangle.

Draw a line that splits the rectangle into two triangles.

That line is a diagonal.

Focus on one of the triangles.

Notice that the triangle is a right triangle.

You will find that the diagonal is the longest line in the triangle.

(The definition of a diagonal is a line that connects two nonconsecutive vertices of a polygon.)(36 votes)

- i have never seen the way he regrouped the numbers while subtracting. (at2:25) how did he do that?(10 votes)
- In American school, this is normally the way of regrouping you'll usually see. You probably already know that the lower math you have, the harder it is to teach, so instead of me TRYING to teach you it, here's a link that can help:https://www.khanacademy.org/math/arithmetic/arith-review-add-subtract/arith-review-subtract-within-100/v/introduction-to-regrouping-borrowing

There you go! :)(7 votes)

- what do you do when you have to find out for B?(4 votes)
- Rearrange the formula (a2+b2=c2) to isolate b: (c2-a2=b2). Now you can solve for b the same way you would for c. :)(6 votes)

- Why cant you get a real value amount for the equation instead of a square root(5 votes)
- When we have perfect squares, then square roots are easy. It is when we have imperfect squares such as 2,3,5 etc. that it gets more complicated. When taking imperfect squares, we end up in a part of numbers called irrational numbers - numbers that cannot be expressed as a fraction of two integers (thus are non-repeating). sqrt(2) is a real number, but it is also an irrational number and thus cannot be expressed in the way you want.

However, for most real world problems, we can round an irrational number (like when we use 3.14 to approximate pi) to assist in calculations. However, anytime that we round, we loose accuracy, so sqrt(2) is more accurate than 1.414, so in Math which is can be more theoretical than real life, we just leave the sqrt(2) alone.(2 votes)

- Would it be possible (if so is it easier?), if we made the equation as W²= (√74)²-7². Because technically it is the same thing right??(5 votes)
- That is technically right, but skipping steps can confuse you later on.(1 vote)

- wait how did he get 5?(3 votes)
- Because 25/5=5(4 votes)

- Why do they always make the videos and why can't they make a written version that isn't just the video's script, is it too much work or something.(3 votes)
- Well, the videos help visual learners more than just a written version, especially with geometry as you really can't get a grasp of some concepts without it being drawn out. In addition to videos, Khan Academy also has some articles scattered throughout the site. To answer your question: Yes, it would be too much work, because the limited KA team is spending effort reproducing already existing content instead of making new content.(4 votes)

- Can you break (√74)² down for me in steps? How do you solve that?(2 votes)
- This is really just a one-step problem. The squaring undoes the process of taking the square root, so the answer is just 74.(5 votes)

- In "Pythagorean theorem word problems" quiz there is an inscribed triangle question giving a result that requires you to find the square root of 4.5√. And find a result of: 2.121. Fair enough, but I can't figure out how to find the square root of a decimal number. Can anyone help me to understand it?(2 votes)
- You could factor the √4.5 as √9 * √(1/2) which leaves you with (3√2)/2 but if you want a decimal answer there is certainly nothing wrong with just using your calculator when presented with something like that.(5 votes)

- In the solution to the following pyramid problem there

is a step, I do not understand.

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pyth-theorem/e/pythagorean-theorem-word-problems?modal=1

L^2 = (31)^2 + (62)^2

L^2 = (31)^2 + (2)^2(31)^2

Why do we have to factor out the 2? The two sides of the triangle containing the slant height are known just like every other triangle?

(2 votes)- That is just a number sense trick to make it easier to do in your head. This way, when you take the square root, you get an exact answer of 31√5 as opposed to a rounded answer if you do it on the calculator.(3 votes)

## Video transcript

A carpet measures 7 feet long
and has a diagonal measurement of square root of 74 feet. Find the width of the carpet. So let's draw ourselves
a carpet here. So let's draw a carpet. It has a length of 7 feet, so
let's say that that is 7 feet, right there. And it's going to be a rectangle
of some kind. So let's say that we're
looking down on the carpet like that. That's our carpet. And then it has a diagonal
measurement of square root of 74 feet. So that means that this
distance, right here-- draw it a little bit neater than that
--this distance right here, the diagonal of the carpet, is
the square root of 74 feet. And what they want to know is
the width of the carpet. Find the width of the carpet. So let's say that this is
the width of the carpet. That is w, right there. Now, you might already realize
that what I have drawn here is a right triangle. Let me make sure
you realize it. This is a 90 degree
angle here. And since that is a triangle
that has a 90 degree angle, it's a right triangle. The side opposite the right
angle, or the 90 degrees, is a hypotenuse, or the
longest side. It is the square root of 74. And the shorter sides
are w and 7. And the Pythagorean Theorem
tells us that the sum of the squares of the shorter side will
be equal to the square of the hypotenuse, so the square
of the longer side. So we get w squared,
this side squared. plus 7 squared, this other side
squared, is going to be equal to the hypotenuse
squared, square root of 74 squared. And then we get w squared plus
49 is equal to the square root of 74 squared. Well, that's just
going to be 74. It is equal to 74. We can subtract 49 from both
sides of this equation. So we have just a w squared
on the left-hand side. Subtract 49 from both sides. The left side-- these guys are
going to cancel out, we're just going to be left with a w
squared --is equal to-- What's 74 minus 49? 74 minus 49, well, we can do a
little bit of regrouping or borrowing here, if we don't
want to do it in our head. We can make this a 14. This becomes a 6. 14 minus 9 is 5. 6 minus 4 is 2. And we have w squared
is equal to 25. So w is going to be equal to
the square root of 25, the positive square root. So let's take the square root
of both sides, the positive square root, and we will
get w is equal to 5. Because we obviously we don't
want it to be negative 5. That wouldn't be a realistic
distance. So the width of the
carpet is 5. And we're done.