Similar Triangles: △ ABG, △ ACF, and △ ADE Lengths of Segments: BG=2 feet, CF=4 feet
Practice makes perfect
We want to identify all pairs of similar triangles in the given diagram, then find the lengths of BG and CF. Let's take a look at the given diagram with the added information from the exercise.
We will first focus on finding all of the similar triangles in the given diagram. We can see that there are three different triangles in the picture, △ ADE, △ ABG, and △ ACF. Notice that all of these triangles share ∠ A.
We found that two angles in △ ABG are congruent to two angles in △ ACF and △ ADE. Because of this, the third angles are also congruent. We can therefore say that all these triangles are similar.
Similar Triangles
△ ABG
△ ACF
△ ADE
Now let's find the lengths of BG and CF. We can use the fact that corresponding angles in similar triangles have equivalent ratios. We will label the length of BG as y and the length of CF as z. Let's take a look at what these ratios look like for our diagram.
AD/DE=AC/CF=AB/BG [0.3em]
⇕ [0.3em]
AD/6=6.32/z=3.16/y
We need to find the length of AD. Let's call this total length x and label these lengths on our diagram.
As we can see in the diagram, AD is created by three the three equal segments AB, BC, and CD. We know that the sum of the lengths of two of these segments is 6.32 feet. This means that 6.32 is 23 the length of AD. Let's write and solve an expression to find the length of AD using x.
We found that the length of AD is 9.48 feet. Let's update our ratios.
AD/6=6.32/z=3.16/y [0.3em]
⇕ [0.3em]
9.48/6=6.32/z=3.16/y
Now we can solve the ratios for the missing values! Let's start with z.
