The Midline Theorem

Two midpoints of a triangle are connected with a segment. Investigate.

1.) What did you observe in your investigation?

2.) What do your observations about the measures of the angles mean? How do they relate to the properties of the segments? 

3.) What did you observe about the lengths of segment DE and segment BC? Was this observation always true?

4.) Based on the discussion, make conjectures about the relationship between segment AB and segment BC.

Pythagorean Theorem

Four right triangles with sides a, b, and hypotenuse c, are arranged to form square GHIJ. You may adjust the size of the triangles by moving point G and their shape by moving point Q. Answer questions 1 and 2 first before moving the slider.

1) What type of quadrilateral is the white region? Justify your answer.

2) What is the area of the quadrilateral in (1)?

3) Move slider R to the extreme right to show slider T. Then, move slider T to the extreme right. What type/s of quadrilateral are the two white regions? Justify your answer.

4.) What are the areas of the quadrilaterals in (3)?

5.) How does the area of the quadrilateral in (1) relate to the areas of the quadrilaterals in (3)?

6.) What equation can we form from the relationship in (4) in terms of a, b, and c?

Task 5 – Distance from Point of Similarity

In the figure below, $latex \triangle ABC \sim A’B’C’$ and point $latex P$ 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 $latex \triangle STU$ and $latex \triangle MNO$.



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, $latex \triangle ABC \sim \triangle A’B’C’$ 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 >>

Task 3 – Connecting Vertices of Similar Triangles

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



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

Task 4 >>

Task 2 – Drawing Similar Triangles (Part 2)

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,

$latex \frac{A’B’}{AB’} = 2$

$latex \frac{A’C’}{AC’} = 2$
 
$latex \frac{B’C’}{BC} = 2$
 

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

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Next: Task 3 of 5

 

Task 1 – Drawing Similar Triangles (Part 1)

In the applet below, given $latex \triangle ABC$, use the Segment tool to draw $latex \triangle A’B’C’$  such that $latex A’B’ = 2AB$, $latex A’B’ = 2AC$, and $latex B’C’ = 2BC$.

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.

In your lesson on Triangle Congruence, you have learned about corresponding angles and corresponding sides. In the two triangles above, the pairs of corresponding angles are $latex A$ and $latex A’$, $latex B$ and $latex B’$, and $latex C$ and $latex C’$. The pairs of corresponding sides are $latex AB$ and $latex A’B’$, $latex BC$ and $latex B’C’$ and $latex AC$ and $latex A’C’$.