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سرفصل: بخش ریاضی / سرفصل: هندسه / درس 7

بخش ریاضی

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• زمان مطالعه 11 دقیقه
• سطح خیلی سخت

دانلود اپلیکیشن «زوم»

این درس را می‌توانید به بهترین شکل و با امکانات عالی در اپلیکیشن «زوم» بخوانید Similar Triangles

Now we can talk about similar triangles. Recall the idea of congruent. If shapes are congruent, that 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 compare them up. The corresponding sides have equal length. The corresponding angles are equal. They are 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, that 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.

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 will automatically create a smaller similar triangle. So GJ, 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 in equal, would be equal in both.

Obviously, they both share the same angle at H. But angle, for example, angle HGJ would have to be equal to angle HFK. It’s not hard to prove that two triangles are similar, using their angles. If just two angles in one triangle are equal to two angles in 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 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 that, that’s two angles.

That’s enough right there to guarantee that those two triangles, the smaller triangle and the larger triangle are similar. 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, these are similar triangles. The ratio of AB over BC that has to equal the ratio of DE over EF. So setting those two ratio equals, that’s the proportion we can setup. Here are the similar triangles again.

Another way to write those 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, then these fractions are greater than one, and this quantity is known as the scale factor. So we use K for scale factor.

So k would be D over E equals EF over BC equals 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. 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 of the smaller triangle, or suppose we connect one of the vertices to the midpoint of the opposite side.

Any length at all that we construct in the smaller triangle, if we construct that exact same length. In the larger triangle we 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’s parallel to the side so we know that we have two similar triangles. FJ, FJ equals 4, that little segment and GH and also JH equals 20. So GH is that opposite side, JH is part of the 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 going to 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 minus EJ.

So that’s 5x minus x, which is 4x. Well now we can solve for x. x equals five and that’s the length of EJ. So, that’s a very easy problem to solve using the idea of scale factor. Now, we’ll talk about similar figures and area.

This is a relatively rare topic that may show up on the more challenging quant questions. So this is not always gonna show up, but this could show up. We know, 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 ha, 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. Pause the video and then we’ll talk about this. So clearly we have two similar triangles.

CDB and CAE, and BD corresponds to 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 a right triangle with legs of 5 and 10. So that area’s 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 will use doubling and halving. Half of 64 is 32. Half of 32 is 16. So we get 16 times 100 is 1,600. 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 minus 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 could 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 better, than not necessarily going to show up, only on more advanced problems.

If all the lengths are multiplied by k, then the area is multiplied by k squared, the scale factor squared.

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