Author: geogebraph

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

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Task 5 >>

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 $latex \triangle EFG$  and name it $latex \triangle HJI$. Connect their corresponding vertices. Did your observation in 2 also hold?

Task 4 >>

In the applet below, $latex A’B’ = 2AB$, $latex A’C’= 2AC$, and $latex B’C’= 2BC$. Use the Segment tool to draw $latex \triangle PQR$ such that $latex PQ = 3AB$, $latex PR= 3AC$, and $latex QR = 3BC$.

Questions

1.) How did you draw ∆PQR?

2.) What do you observe about the corresponding angles of the ∆ABC, ∆A’B’C’ and ∆PQR?

3.) Based on Task 1 and Task 2, make a conjecture about your observations above.

Discussion

In Task 1 and Task 2, you have observed that it is easier to draw similar triangles if you make the corresponding sides of the triangles parallel. In addition, you have also observed that the corresponding angles of the triangles are congruent. For example,

$latex \angle A \cong \angle A’$

$latex \angle B \cong \angle B’$

and

$latex \angle C \cong \angle C’$

In drawing the triangles, you made sure that the ratio of the lengths of corresponding sides is equal. In other words, you made the length of the corresponding sides proportional. For instance, the ratio of the lengths of the corresponding sides of to those of is 2. That is,

$latex \frac{A’B’}{AB’} = 2$

Two triangles whose corresponding angles are congruent are called similar triangles. The lengths of the corresponding sides of similar triangles are proportional.

Questions

1.) How did you draw $latex \triangle A’B’C’$?

2.) What do you observe about the angles of the two triangles?

3.) Draw other triangles satisfying the same conditions and see if your observations still hold.