Problem Statement
Construct a triangle with sides 5 cm, 5.4 cm, and 6 cm. Then construct a similar triangle whose sides are \(\frac{3}{4}\) times the sides of the original triangle.
Mathematical Foundation
For similar triangles, the ratio of corresponding sides is constant:
\[
\frac{A'B'}{AB} = \frac{B'C'}{BC} = \frac{A'C'}{AC} = k
\]
where \(k = \frac{3}{4}\) in this case.
The sides of the new triangle will be:
\[
\begin{align*}
5 \times \frac{3}{4} &= 3.75 \text{ cm} \\
5.4 \times \frac{3}{4} &= 4.05 \text{ cm} \\
6 \times \frac{3}{4} &= 4.5 \text{ cm}
\end{align*}
\]
Step-by-Step Construction Process
Part 1: Constructing the Original Triangle
1
Draw the base: Draw segment AB = 6 cm
2
Construct point C using compass: With A as center, radius 5 cm, draw an arc. With B as center, radius 5.4 cm, draw another arc. The intersection point is C.
3
Complete the triangle: Join AC and BC to form triangle ABC.
Part 2: Constructing the Similar Triangle
Method: Dilation from a Point
4
Choose a center of dilation: Select point A as the center.
5
Draw rays: Draw ray from A through B and ray from A through C.
6
Mark points for \(\frac{3}{4}\) ratio: On ray AB, mark point B' such that AB' = \(\frac{3}{4}\) × AB = 4.5 cm. On ray AC, mark point C' such that AC' = \(\frac{3}{4}\) × AC = 3.75 cm.
7
Complete the similar triangle: Join B'C' to form triangle AB'C'.
Verification
Original Triangle Sides
| Side |
Length |
| AB |
6.00 cm |
| BC |
5.40 cm |
| AC |
5.00 cm |
Similar Triangle Sides
| Side |
Length |
| A'B' |
4.50 cm (6 × 3/4) |
| B'C' |
4.05 cm (5.4 × 3/4) |
| A'C' |
3.75 cm (5 × 3/4) |
Geometric Proof of Similarity
The triangles are similar by Side-Side-Side (SSS) Similarity:
\[
\frac{A'B'}{AB} = \frac{B'C'}{BC} = \frac{A'C'}{AC} = \frac{3}{4}
\]
Interactive Features
In the Geogebra applet, you can:
Adjust the scale factor using the slider
Move vertices of the original triangle
Observe how the similar triangle maintains the ratio
Verify measurements in real-time
