Identifying values in scale drawings

We're told that figure A is a scale image of figure B. So that's figure A; this is figure B. Here, the scale that maps figure A to figure B is one to two and one half. What is the value of x?

All right, pause this video and see if you can figure it out. All right, so x is the length of this side right over here in figure A, and the corresponding side on figure B has length five.

One way to think about it is the ratio between x and five should be one to two and a half because that's the scale that goes from figure A to figure B. Figure A to figure B. So the ratio—let me write this out—the ratio between x and 5. So the ratio of x to 5, this should be an equivalent ratio as 1, one to two and a half.

And that one is hard to read; let me make a deeper blue. There you go! All right, so let's just think about how to do this. To go from two and a half to five, to go that way, you would multiply by two. So, to go from one to x, you would also multiply by two.

So the value of x is equal to two. If these are scaled up, you multiply this by two and a half, you get to five. So the scale factor is one to two and a half.

Let's do another example. We're told figure A is a scale image of figure B, and we see them both right over here. Once again, we need to figure out what x is going to be, and they don't give us the scale factor, but we can figure out the scale factor.

How do we do that? Well, we can see when you go from... of course, this side right over here corresponds to this side. It's the shorter side that forms a right angle with the base, and this is the longer side that forms a right angle with the base.

So you could set up some ratios. You could say look, the ratio of 3 to 4, you could say the ratio of 3 to 4. Let me do that blue color; the ratio of 3 to 4 is going to be the same thing as the ratio of x to 7.2. The ratio of x to 7.2, x to 7.2.

And so how do we figure out what x is going to be? Well, how do you go from 4 to 7.2? What do you have to multiply by? You might want to get out a calculator, or you might be able to do this in your head.

4 times 18 is 72, which is 2 times 36, which is 2 times 18. So 4 times 18 would be 72, or 4 times 1.8 would be 7.2. If you don't feel good about that mental arithmetic, you could just do the division.

4 goes into 7 one time. 1 times 4 is 4; subtract, you get a 3. You're gonna have your decimal right over there. Bring down the two. Thirty-two—4 goes into 32 eight times; 8 times 4 is 32, and we're done.

So to go from four to seven point two, you have to multiply by one point eight. And so to go from three to x, you also have to multiply by 1.8. And so x is 3 times 1.8.

What is that going to be? Well, 3 times 18 is... what? 30? It's 54. So this is going to be... let me verify that or show you that. 18 or 1.8 times 3.

3 times 8 is 24. 3 times 1 is 3 plus 2 is 5. One number behind the decimal point, 5.4, and we're done.