Unit 3, Lesson 13
Using the Pythagorean Theorem and Similarity
Geometry
13.1
Warm-up
Instructional Routines
NoneMaterials
To Gather
Four-function calculatorsActivity Narrative
The goal of this activity is to get students familiar with the two smaller right triangles formed by drawing an altitude to the hypotenuse of a right triangle. The activity previews the activities that students will do later in this lesson and the next. Listen to hear if students compare the two smaller triangles to the larger triangle, or make conjectures about what other proportional relationships might be present.
Launch
NoneActivity
NoneIs triangle similar to triangle ? Explain or show your reasoning.
Student Response
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Building on Student Thinking
If students struggle to see which angles and sides correspond, encourage them to copy each triangle onto tracing paper and rotate them so they all have the same orientation. Colored pencils can also help students identify corresponding parts.
Activity Synthesis
Ask students how many triangles they see in the diagram. Ask them how many of the triangles are similar. If students aren’t sure, they will have more opportunities later in the lesson to untangle the diagram.
Tell students that an altitude in a triangle is a line segment from a vertex to the opposite side such that the segment is perpendicular to that side.
Lesson Synthesis
Ask students to find the measures of all of the missing angles and to justify how they know. (Angle measures by the Triangle Angle Sum Theorem. Angle measures because it is the complement of angle . Angle measures by the Triangle Angle Sum Theorem.)
Repeat with a variable in place of the angle. (Angle measures . Angle measures . Angle measures .) Once the angles are calculated, display the triangles redrawn so that they are all oriented the same way. Invite multiple students to explain why the triangles formed by drawing the altitude to the hypotenuse must be similar. Encourage students to generalize their conclusions to any right triangle with the altitude drawn to the hypotenuse. Ask students to list all of the equivalent ratios that they can among the three similar triangles. Point out again the equivalent ratios in which one side appears twice in the same equation. Students will need to use these in a subsequent lesson to prove the Pythagorean Theorem.
Remind students that these triangles are no different than any other set of similar triangles. All the same strategies they have for reasoning about similar triangles still apply to this diagram.
Student Lesson Summary
When we draw an altitude from the hypotenuse of a right triangle, we get three pairs of similar triangles that can be used to find missing lengths. An altitude is a segment from one vertex of the triangle to the line containing the opposite side and that is perpendicular to the opposite side. For right triangle we can draw the altitude .
Why are triangles , , and all similar to each other?
Triangles and are similar by the Angle-Angle Triangle Similarity Theorem because angle is in both triangles, and both triangles are right triangles, so angles and are congruent. Triangles and are similar by the Angle-Angle Triangle Similarity Theorem because angle is in both triangles, and both triangles are right triangles, so angles and are congruent. Because triangles and are both similar to triangle , they are also similar to each other.
Because the triangles , , and are all similar, corresponding angles are congruent and pairs of corresponding sides are scaled copies of each other, by the same scale factor. We can use the proportionality of pairs of corresponding side lengths to find missing side lengths. For example, suppose we need to find and know that and . Because triangle is similar to triangle , we know that . So , and . Or, suppose we need to find and know that and . Because triangle is similar to triangle , we know that . So , and .