What is meant when someone says similar triangles? The next video in our Geometry series addresses this question, and more.
Transcript of Video: Similar Triangles
Now we can talk about similar triangles. Recall the idea of congruence. If shapes are congruent, it means they have the same shape and the same size. So for example, these are two congruent shapes. They don’t have to be in the same orientation. We can flip them around, we can rotate them, but these are essentially the same shape and so we can pair them up.
The corresponding sides have equal length, the corresponding angles are equal, they’re exactly the same shape just in different orientations. Sometimes, in geometry and in real life, we need to examine two figures that have the same shape, but different sizes. Two geometric figures with the same shape and different sizes are called similar. And so what we have here, those two shapes, it is the same shape.
The second one is a smaller copy of the first one, and again it’s in a different orientation. It’s been flipped around and rotated a little bit, but that is in fact the same shape.
Similar Triangles: Angles
So the first thing we’ll talk about are angles in similar figures. The first big idea of similar figures is that similar figures have equal angles.
So B corresponds to J. So angle B has to equal angle J, angle C has to equal angle K, and angle D has to equal angle L. And so each angle is equal to its corresponding angle in the similar triangle. Notice that, if a segment goes across a triangle and is parallel to one of the sides, it automatically creates a smaller similar triangle.
So if GJ is parallel to FK, well that means that triangle HFK (the larger triangle), is similar to triangle HGJ (the smaller triangle). And it also means that the angles would be equal in both. Obviously they both share the same angle at H, but for example angle HGJ would have to be equal to angle HFK.
It’s not hard to prove that two triangles is similar using their angles. If just two angles and one triangle are equal to two angles and another triangle, that’s enough to establish that the two triangles are similar. So for example, here we know that the smaller triangle has a right angle, the larger triangle also has a right angle. So that’s one thing that they have in common.
And they share that angle at P. And so that’s another angle that, since it’s common to both of them, they share those two angles, that’s enough right there to guarantee that those two triangles, the smaller triangle and the larger triangle are similar.
Similar Triangles: Sides
Now we can talk about sides in similar figures. The sides in similar figures are proportional.
Well what does that mean? This means that we can set up proportions. The ratio of any two sides of one triangle has to be equal to the ratio of the corresponding sides in the other triangle. So for example, if these are similar triangles, the ratio of AB over BC that has to equal the ratio of DE over EF.
So setting these two ratios equal, that’s the proportion we can set up. Here are the similar triangles again. Another way to write these is in the form of side of one triangle, over the corresponding side of the other triangle. So we could write AB over DE. And the advantage of this is now we can create an equation that relates all the sides of both triangles.
So AB over DE, those are corresponding sides. BC over EF those are corresponding sides and AC over DF, those are corresponding sides. So all three of those ratios have to be equal. If the sides of the bigger triangle are written in the numerator, and these fractions are greater than one, and this quantity is known as a scale factor.
So we use K for scale factor. So K would be D over E = EF over BC = DF over AC. If we know any length in the smaller triangle the corresponding length in the larger triangle is just the original length times the scale factor. And in fact the amazing thing is not just for the sides of the triangle, but any length at all.
So for example, suppose we construct an altitude from the smaller triangle, or suppose we connect one of the vertices to the midpoint of the opposite side. So take any length at all that we construct in the smaller triangle. If we construct that exact same length in the larger triangle–we’d multiply the length in the smaller triangle by the scale factor and it would be the length in the larger triangle. So this is an incredibly powerful tool scale factor.
Here’s a practice problem, pause the video and then we’ll talk about this.
Okay, so we have a segment that is parallel to the side so we know that we have two similar triangles. FJ = 4, that little segment, and GH and also JH = 20. So GH is that opposite side and JH is part of that side and we want to find the length EJ.
Well, first of all let’s compare the two sides that are obviously corresponding sides. So we take the ratio of GH over FJ. This is gonna be the scale factor and of course this is 20 over 4, so this is 5. So we don’t know EJ. So let’s just call that x.
That’s the thing we’re looking for, we don’t know it. We do know that EH would have to be 5x, it would have to be 5 times that length. Well, the number that we’re actually given is JH. Well, notice that JH, that’s 20. Of course that has to equal EH- EJ. So that’s 5x- x, which is 4x.
Well now we can solve for X. X = 5 and that’s the length of EJ. That’s a very easy problem to solve, using the idea of scale factor. Now, we’ll talk about similar figures and area.
Rare Topic — But Could Be Tested
This is a relatively rare topic that may show up on more challenging questions.
So this is not always going to show up, but this could show up. If we know that every length in one shape gets multiplied by a scale factor to create another shape, then what is the relationship between the areas of the two shapes? Let’s think about this for triangles. We know that the area of a triangle is given by area equals one-half base times height.
Suppose each length increases by a scale factor k. Then b is multiplied by k, And h is also multiplied by k. That’s two factors of k. This means that the area is multiplied by two factors of k, in other words, k squared. When lengths are multiplied by k, area is multiplied by k squared.
Here’s a practice problem. Pause the video and then we’ll talk about this.
So clearly the scale factor is the ratio of 30 divided by 6, which is 5. The scale factor is 5, the scale factor squared is 25. The area is multiplied by 25. So the small one has an area of 12, the large one must have an area of 25 times 12.
For that, we’ll use the doubling and halving trick. Double 25 to 50 halve 12 to 6, do it again. Actually, we don’t even need to do it again. 6 times 50 is 300, that’s pretty easy to see. So that’s the area of the larger triangle. Here’s a harder practice problem.
Harder Practice Problem
Pause the video and then we’ll talk about this.
So clearly we have two similar triangles CDB and CAE, and BD corresponds the side AE. So we set up the ratio AE over BD, this will give us the scale factor. This is 40 divided by 5 which is 8, that’s the scale factor. So the ratio of the areas is gonna be the scale factor squared.
Well we can easily find the area of that small triangle, it’s just the right triangle with legs of 5 and 10. So that area is gonna be one-half, 5 times 10, that’s gonna be 25. Well the area of the larger triangle is going to be 64 times 25, scale factor squared times the smaller area. For this, we’ll use doubling and halving, half of 64 is 32.
Half of 32 is 16, so we get 16 times 100 is 1600. That’s the area of the large triangle, and of course the shaded area is just large triangle minus small triangle. And so that’s gonna be 1600- 25, which is 1575. That’s the area of the shaded region. In summary, similar figures have the same shape but different sizes.
They’re like expanded or shrunken down versions of the same shape. The angles in similar figures are always equal. We can prove two triangles are similar if they simply share two angles. Sides in similar figures are proportional, there’s a number of ways we can set up those proportions. The scale factor, k, is the factor by which all lengths in the smaller figure were multiplied to arrive at the lengths in the larger figure.
And finally, and this is the fact that may show up that it’s not necessarily gonna show up, only on more advanced problems. If all the lengths are multiplied by k, then the area is multiplied by the k². The scale factor squared.