## Task 5 – Distance from Point of Similarity

In the figure below, $\triangle ABC \sim A'B'C'$ and point $P$ is the point of similarity of the two triangles.

1.) What is the ratio of the lengths of the corresponding sides?

2.) What is the ratio of the distance from point P to the corresponding vertices?

3.) What conjecture can you make from your answer in 1 and 2? Verify your conjectures by drawing a different pair of similar triangles.

Explanation

The ratio of the corresponding sides of two similar triangles is the same as the ratio of the distance of their corresponding vertices to the center of similarity. The ratio of the corresponding sides of two similar triangles is the same as the ratio of the distance of their corresponding vertices. In the figure above, ∆ABC ~ ∆A’B’C’ so

and

## Task 4 – Point of Similarity

Shown below are $\triangle STU$ and $\triangle MNO$.

Questions

1.) Are the two triangles similar? Why?

2.) What is the ratio of the lengths of their corresponding sides?

3.) Use the Line tool to connect the corresponding vertices of the two triangles. Did your observations in Task 2 and Task 3 hold?

4.) Make a conjecture about the lines connecting the corresponding vertices of similar triangles.

[toggle title=’Explanation’ title_font_size=’14px’]If two triangles are similar and their corresponding sides are parallel, then the line connecting their corresponding vertices will intersect at a point. This point is called the center of similarity or point of similarity.

In Task 3, $\triangle ABC \sim \triangle A'B'C'$ and their corresponding sides are parallel. The three lines connecting the corresponding vertices intersect point $P$ as shown above. In Task 4, $\triangle STU \sim \triangle MNO$, but their corresponding sides are not parallel, so the three lines connecting their corresponding vertices do not intersect at a point.

[/toggle]

## Task 3 – Connecting Vertices of Similar Triangles

Shown in the applet below are $\triangle EFG$ and $RST$.

Questions

1.) Are the two triangles similar? Why?

2.) Use the Line tool to connect the corresponding vertices of the two triangles. What do you observe about the lines?

3.) Draw another triangle that is similar to $\triangle EFG$  and name it $\triangle HJI$. Connect their corresponding vertices. Did your observation in 2 also hold?