Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles
Use the diagram below to complete parts (a)–(c).
a. Are the triangles shown above similar? Explain.
b. Determine the unknown lengths of the triangles.
c. Explain how you found the lengths in part (a).
Recall that an altitude of a triangle is a perpendicular line segment from a vertex to the
line determined by the opposite side. In △ 𝐴𝐵𝐶 to the right, 𝐵𝐷 is the altitude from
vertex 𝐵 to the line containing 𝐴𝐶.
a. How many triangles do you see in the figure?
b. Identify the three triangles by name.
We want to consider the altitude of a right triangle from the right angle to the hypotenuse. The altitude of a right triangle splits the triangle into two right triangles, each of which shares a common acute angle with the original triangle.
In △ 𝐴𝐵𝐶, the altitude 𝐵𝐷 divides the right triangle into two sub-triangles, △ 𝐵𝐷𝐶 and △ 𝐴𝐷𝐵.
c. Is △ 𝐴𝐵𝐶~ △ 𝐵𝐷𝐶? Is △ 𝐴𝐵𝐶~ △ 𝐴𝐷𝐵? Explain.
d. Is △ 𝐴𝐵𝐶 ~ △ 𝐷𝐵𝐶? Explain.
e. Since △ 𝐴𝐵𝐶 ~ △ 𝐵𝐷𝐶 and △ 𝐴𝐵𝐶~ △ 𝐴𝐷𝐵, can we conclude that △ 𝐵𝐷𝐶 ~ △ 𝐴𝐷𝐵? Explain.
f. Identify the altitude drawn in △ 𝐸𝐹𝐺.
g. As before, the altitude divides the triangle into two sub-triangles, resulting in a total of three triangles including the given triangle. Identify them by name so that the corresponding angles match up.
h. Does the altitude divide △ 𝐸𝐹𝐺 into two similar sub-triangles as the altitude did with △ 𝐴𝐵𝐶? The fact that the altitude drawn from the right angle of a right triangle divides the triangle into two similar sub-triangles, which are also similar to the original triangle, allows us to determine the unknown lengths of right triangles.
Consider the right triangle △ 𝐴𝐵𝐶 below.
Draw the altitude 𝐵𝐷 from vertex 𝐵 to the line containing 𝐴𝐶. Label 𝐴𝐷 as 𝑥, 𝐷𝐶 as 𝑦, and 𝐵𝐷 as 𝑧.
Find the values of 𝑥, 𝑦, and 𝑧.
Now we will look at a different strategy for determining the lengths of 𝑥, 𝑦, and 𝑧. The strategy requires that we complete a table of ratios that compares different parts of each triangle.
Make a table of ratios for each triangle that relates the sides listed in the column headers.
Our work in Example 1 showed us that △ 𝐴𝐵𝐶 ~ △ 𝐴𝐷𝐵 ~ △ 𝐶𝐷𝐵. Since the triangles are similar, the ratios of their
corresponding sides are equal. For example, we can find the length of 𝑥 by equating the values of shorter leg:
hypotenuse ratios of △ 𝐴𝐵𝐶 and △ 𝐴𝐷𝐵.
Why can we use these ratios to determine the length of 𝑥?
Which ratios can we use to determine the length of 𝑦?
Use ratios to determine the length of 𝑧.
Since corresponding ratios within similar triangles are equal, we can solve for any unknown side length by equating the values of the corresponding ratios. In the coming lessons, we will learn about more useful ratios for determining unknown side lengths of right triangles.
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