Lesson 13

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.

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.

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 an altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.

In a subsequent lesson, students will use proportional relationships in a similar diagram to prove the Pythagorean Theorem. It’s important for students to practice seeing the relationships between the triangles formed by drawing an altitude to the hypotenuse of a right triangle. Monitor for students who:

• use the Pythagorean Theorem to find the third side length
• use the scale factor they determined to find the third side length
• use equivalent ratios to find the third side length

If students struggle to find the similarity statement, have them also trace the large triangle without the altitude and then have them orient all three triangles in the same direction.

Invite a student to share who:

• used the Pythagorean Theorem to find the third side length
• used the scale factor they determined to find the third side length
• used equivalent ratios to find the third side length

If no student used equivalent ratios, ask students to write several equivalent ratios for the three triangles. In a subsequent lesson, students will need to recognize, write, and manipulate equivalent ratios involving the same side lengths.

In this activity, the angle measures are not given, so students need to convince themselves that the triangles in question are similar. Students may be tempted to assume they are similar simply because of the triangles in the previous diagram that looked like this, so monitor for students actively trying to convince themselves that the Angle-Angle Triangle Similarity Theorem applies in this case. Look for:

• students measuring angles (with tracing paper or protractors)
• students measuring one or two angles and calculating the rest
• students using algebraic or logical reasoning to convince themselves that the angles would be congruent

Students also get another chance to write equivalent ratios. One key thing they might notice is that a lot of the sides are in more than one ratio.

If students assume the triangles are similar because the diagram looks like the previous diagrams, ask them how they know.

Invite students to explain how they convinced themselves that triangles \(ABC, ACD, \) and \(CBD\) are similar. Select students to share their thinking in this order:

• students who used tracing paper or measured all the angles 
• students who measured just one or two angles and calculated the rest
• students who used algebraic or logical reasoning to convince themselves that the angles would be congruent

After students who traced or measured share, make sure all students understand and appreciate the idea that it’s necessary to know that the triangles are similar before considering missing side lengths and that finding angle measures is important to establish the triangles are similar.

Finally, ask students to share equivalent ratios that they found. Record equivalent ratios in which the same side appears twice in a separate column, for example \(\frac{AD}{DC} = \frac{DC}{DB}\) and \(\frac{CB}{AB} = \frac{DB}{CB}\). Ask students what they notice about the equivalent ratios grouped together.