In previous lessons, students reasoned about dilations and confirmed their properties. In this lesson, students use the definition of dilation to prove that the triangles formed by connecting midpoints of two sides are dilations of the original triangle. Then they deduce properties of these segments using a new line of argument: proving that two figures are dilations of one another is a way to prove the figures have properties of dilations (MP3).
This prepares students to prove that pairs of triangles with certain properties must be dilations of one another, and therefore must be similar triangles. Note that in a later lesson, students will prove the converse of the theorems they explore in this lesson - that a segment parallel to one side of a triangle divides the other two sides proportionally. If students notice or wonder about this fact, leave it as a conjecture for now.
- Justify that a segment dividing two sides of a triangle proportionally is parallel to the third side.
- Prove that a segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side (orally and in writing).
- Let’s draw segments connecting midpoints of the sides of triangles.
- I can explain why the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half the length of the third side.
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.