Dilating Lines and Angles and Splitting Triangle Sides

​Dilating Lines and Angles Splitting Triangle Sides with Dilation, Part 1

​Learning Goals:

• ​Let’s dilate lines and angles.

• Let’s draw segments connecting midpoints of the sides of triangles.​

Open Ended

Triangle A’B’C’ is a dilation of triangle ABC using center P and scale factor 2.

What do you think is true about the angles in A’B’C’ compared to the angles in ABC? And do you think it would be true for angles in any dilation?

Open Ended

If I dilate point A using center C and scale factor ¾ , and dilate point B using center C and scale factor ⅓, what happens to each of the points?

Open Ended

If I dilate point point  D using center C and scale factor 3/2., and line CE using center C and scale factor 2., what happens to point  D and line CE with respect to point C

Open Ended

What happens when the center of dilation is on a line and then you dilate the line?

Jada claims that all the segments in ABC are parallel to the corresponding segments in A’B’C’.

How could we write Jada's claim as a conjecture.

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​Proof in Parallel

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Draw

Jada dilated triangle ABC using center  and scale factor 2.

Jada claims that all the segments in ABC are parallel to the corresponding segments in A’B’C’. Can we write Jada's claim as a conjecture.

​Proofs in Parallel (Theorem)

• because we know the figures are dilations, we know that corresponding angles are congruent

• because we know that corresponding angles are congruent, we know that the lines cut by the transversal are parallel

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A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (Theorem)

Open Ended

In Jada’s diagram the scale factor was greater than one. Would your proof have to change if the scale factor was less than one?

Open Ended

Here’s a triangle ABC with midpoints L, M, and N.

What do you notice? What do you wonder?

Open Ended

If were to state that triangle ABC is a dilation of triangle AMN. What is the center of the dilation? What would be the scale factor?

See if you and your elbow partner agree.

Open Ended

Based upon my previous statement, can I show that segment BC is twice as long as segment MN? Explain your reasoning

Theorem:

If a line divides two sides of a triangle proportionally, the line must be parallel to the third side of the triangle.

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​Splitting Triangle Sides with Dilation

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Open Ended

Here’s a triangle ABC. M is ⅔  of the way from A to B. N is ⅔  of the way from A to C.

What can you say about segment MN, compared to segment BC? Explain your reasoning.

​Dilating Lines and Angles Splitting Triangle Sides with Dilation, Part 1

​Learning Goals:

• ​Let’s dilate lines and angles.

• Let’s draw segments connecting midpoints of the sides of triangles.​