Properties of Similar Triangles
In this video, I want to take a brief detour from algebra to look at a geometry application for what we just talked about, meaning the topic of equivalent ratios. Now, in geometry, you have a property of triangles called “similar triangles”.
When you have two triangles that are similar, you know that the ratios of their corresponding sides are equal to one another. You have a multiple equivalent ratio problem. Let’s say we have two triangles. Let’s say this first triangle—this isn’t going to be drawn to scale—but let’s say we have triangle ABC and triangle XYZ.
These two triangles, we’re told that they’re similar. What that tells us is that side A, the ratio of side A to side X is the same as the ratio of side B to side Y, which is the same as the ratio of the side C to side Z. If we’re given the lengths of the sides on one of the two triangles and then the length of any one side of the triangle, we can find the missing quantities.
Let’s say we’re told that triangle ABC has sides of length 4, 3, and 2. Then, we’re told that side X is equal to 12. We have these four quantities and we need to find the remaining two. We can use equivalent ratios to find their values.
We have the value for X, so this will be the ratio we use to find the exact value of the ratio. A is equal to 4. X is equal to 12. This ratio is equal, is the same as, 1/3. B and Y are going to have a ratio of one to three and C and Z are going to have a ratio of one to three.
We can set up two more of these equivalent fraction problems to find the values for Y and Z. B/Y, or 3/Y, is going to be equal to 1/3. Now we can cross multiply to find the value of Y. We get 3 times 3 is 9 and Y times 1 is Y.
The value of Y is 9. We can do the same thing now with C and Z. We know the value of C is 2. 2/Z is equal to 1/3. When we cross multiply, we get Z equals 6. This is just one example of an application of equivalent ratios in geometry.