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Course: High school geometry>Unit 4

Lesson 7: Solving modeling problems with similar & congruent triangles

Similarity FAQ

What is the relationship between similarity and geometric transformations?

Similarity is about two shapes having the same proportions. Two shapes can be similar even if one is bigger or smaller than the other, as long as they keep the same proportions.
In terms of transformations, we can use four different types to create similar shapes:
• Dilations: enlarging or shrinking a shape
• Translations: moving a shape from one location to another
• Rotations: rotating a shape around a point
• Reflections: flipping a shape over a line
We can use any combination of these transformations to create shapes that are similar to one another.
Try it yourself with our Similarity & transformations exercise.

How do we determine that two triangles are similar?

Similar triangles have the same shape, but they might not have the same size. To determine if two triangles are similar, we need to look at three things:
• Side lengths: If the ratios of corresponding side lengths are equivalent, the triangles are similar. For example, if $\mathrm{△}ABC$ has side lengths of $6$, $8$, and $10$, and $\mathrm{△}DEF$ has side lengths of $9$, $12$, and $15$, we can find that the ratios are equal:
$\frac{6}{9}=\frac{8}{12}=\frac{10}{15}$.
• Angle measurements: If the corresponding angles have equal measures, the triangles are similar. For example, if $\mathrm{△}ABC$ has angles measuring $30\mathrm{°}$, $60\mathrm{°}$, and $90\mathrm{°}$, and $\mathrm{△}DEF$ also has angles measuring $30\mathrm{°}$, $60\mathrm{°}$, and $90\mathrm{°}$, the two triangles are similar. Conveniently, if we know two angle measures of a triangle, we can calculate the third. So if $2$ corresponding angle pairs have equal measures, we know the third corresponding angle pair also has equal measures.
• Side-Angle-Side: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, the triangles are similar. For example, if $\mathrm{△}ABC$ has sides that measure $6$ and $8$ and an included angle of $60\mathrm{°}$, and $\mathrm{△}DEF$ has sides that measure $9$ and $12$ and an included angle of $60\mathrm{°}$, the two triangles are similar.

How does self-similarity work in right triangles?

When we draw an altitude to the hypotenuse of a right triangle, it creates two smaller right triangles. Interestingly, these two smaller right triangles are both similar to the original right triangle and to each other.
This happens because the altitude divides the hypotenuse into two segments, but also creates two new right angles. The two new right angles, along with the two segments, give us two right triangles that share the same proportions as the original right triangle.
This self-similarity is useful in geometry, as it allows us to use properties of similar triangles to solve problems. For example, if we know the length of the hypotenuse and one leg of the original right triangle, we can use this self-similarity to find the lengths of the two segments created by the altitude.
Try it yourself with our Use similar triangles exercise.

What can we prove with triangle similarity?

One thing we can prove using triangle similarity is the Pythagorean theorem. For example, consider a right triangle with sides $a$, $b$, and $c$, where $c$ is the hypotenuse. Divide the triangle into two smaller, similar right triangles by drawing a perpendicular from the right angle to the hypotenuse.
Because the triangles are similar, we can write ratios of corresponding side lengths, which ultimately simplifies to ${a}^{2}+{b}^{2}={c}^{2}$, the Pythagorean theorem!
We can use triangle similarity to show that the three medians of a triangle intersect at a single point. We can use triangle similarity to prove that the three altitudes of a triangle also intersect at a single point.
Another interesting property we can derive from triangle similarity is that when two chords (line segments with both endpoints on a circle) intersect, the product of the two segments on one chord is equal to the product of the two segments on the other chord.
There are many more properties we can prove using triangle similarity, and these are just a few examples.

Want to join the conversation?

• For anyone who didn't understand something, feel free to ask below, I'll try to help :)
• this dont make sense
• exactly just understand
• how do i find the units ?
• that depends on the problem you answer
• makes no sense :)
• Then you should start from the beginning :D