In this video, we're looking at similar shapes, which are shapes like these two, that have exactly the same shape, but are just different sizes. You can normally tell that two shapes are similar just by looking at them, and seeing if they're the same shape. But to be certain, you'll need to know all of the angles inside of each shape. For example, if we fill in all of these angles, you can see that both shapes have exactly the same angles.
And this tells us for certain... that these two shapes are similar. Now when it comes to exams they'll normally just tell you that the two shapes are similar and instead they'll ask you to work out an unknown side on one of the shapes. For example if we look at this question here we're told that these two shapes are similar and we're being asked to find a missing side length x. The key to questions like is to find the scale factor between the two shapes, which is basically how many times bigger the larger shape is than the smaller shape.
To find it we just compare two equivalent sides, like these two bottom sides, and find how many times bigger the larger one is than the smaller one. So because 6 centimeters is 2 times bigger than 3 centimeters, the scale factor here is 2. And if you didn't know that, you could have just divided the larger length of 6 by the smaller length of 3, and that would give you 2, which tells you that 6 is 2 times bigger than 3. A scale factor of 2 means that every length on our larger shape will be 2 times bigger than the equivalent side on our smaller shape. So if we're looking for x, and we know that its equivalent side on the smaller shape is this 5cm side here, then all we have to do is multiply the 5cm by 2, to find that side x must be 10cm long. Let's have a go at another one. So this time we're told that these two triangles are mathematically similar, and we're being asked to work out the lengths of the missing sides x and y.
First of all, the term mathematically similar just means similar, like we've been talking about throughout this video. So whenever you hear the term similar or mathematically similar in a maths paper, they both mean exactly the same thing. Now just like before, the first thing we're going to want to do here is find the scale factor between the two shapes. So because we're given the bottom lengths of these two triangles, we should compare the 8cm side and the 12cm side to figure out how many times bigger the 12cm one is than the 8cm one. And remember that to do that, we just do 12 divided by 8, which is 1.5, meaning that our scale factor is 1.5.
So we could say that D is an enlargement of shape C by a scale factor 1.5. Now that we know the scale factor, we can start to work out the missing sides. So to find the length of X, we just look at the equivalent side on the smaller shape, which is this 26 centimeter one, and then we can multiply 26 centimeters by 1.5 to find that X. must be 39cm long.
To find the length of y is a bit different, because y is on the smaller shape. So once we've found its equivalent side, which is this 45cm one, we're gonna have to divide it by the scale factor of 1.5, rather than multiply it. So because 45 divided by 1.5 is 30, y must be 30cm long. If you ever forget whether you're meant to multiply or divide by the scale factor, Just look at the two shapes and think about what's sensible. For example here, when trying to find y using the 45cm side, we're going from a larger shape to a smaller shape.
So we're of course going to have to divide by the scale factor so that we get a smaller value. One last thing I want to add before we finish is that it's much easier if you always find the scale factor from the smaller shape to the bigger shape, like we did in this question. You can also find it from the bigger shape to the smaller shape, but you'll end up with a scale factor that's less than one, which is confusing to work with.
For example, if we wanted to find the scale factor from d to c, we'd have to do eight centimeters divided by 12 centimeters, which is two thirds. So basically c is two thirds as big as d. And this is going to be harder for us to work with, because it's harder to multiply and divide by a number smaller than one.
So just remember to always find the scale factor from the smaller shape to the bigger shape and work with that one. Anyway, that's everything for this video. If you want to practice any questions on this stuff or anything else in science or maths, then head over to our resource website by clicking the link in the top right corner of this screen.
And otherwise, we'll see you next time. Thanks!