Question

Construct an equilateral triangle with each side 5 cm. Then construct another triangle whose sides are `2/3` times the corresponding sides of ΔАВС.

Answer 1

\[ \begin{array}{r l} \textbf{Given:} & \text{An equilateral triangle } ABC \text{ with each side } 5 \text{ cm, to be reduced by the scale factor } \dfrac{2}{3}. \\[4pt] \textbf{To Find:} & \text{The construction of a triangle similar to } ABC \text{ with each side } \dfrac{2}{3} \text{ of the corresponding side, and the lengths of its sides.} \\[4pt] \textbf{Solution:} & \text{Step 1: Draw } BC = 5 \text{ cm; with centre } B \text{ and centre } C, \text{ each with radius } 5 \text{ cm, draw arcs meeting at } A, \text{ and join } AB \text{ and } AC. \\[4pt] & \text{Step 2: Draw a ray } BX \text{ making an acute angle with } BC \text{ on the side opposite to } A. \\[4pt] & \text{Step 3: Since the greater of } 2 \text{ and } 3 \text{ is } 3, \text{ mark three points } P, Q, R \text{ on } BX \text{ such that } BP = PQ = QR. \\[4pt] & \text{Step 4: Join } RC \text{ and through } Q, \text{ the second point, draw a line parallel to } RC, \text{ meeting } BC \text{ at } C'. \\[4pt] & \text{Step 5: Through } C' \text{ draw a line parallel to } CA, \text{ meeting } BA \text{ at } A'; \text{ then } A'BC' \text{ is the required triangle.} \\[4pt] & \text{Justification: in triangle } BRC, \text{ since } QC' \text{ is parallel to } RC, \text{ the Basic Proportionality Theorem gives } \dfrac{BC'}{BC} = \dfrac{BQ}{BR} = \dfrac{2}{3}. \\[4pt] & \text{Since } C'A' \text{ is parallel to } CA, \text{ the triangles } A'BC' \text{ and } ABC \text{ are similar by the AA criterion, so their sides are proportional:} \\[4pt] & \begin{aligned} \frac{A'B}{AB} = \frac{BC'}{BC} = \frac{A'C'}{AC} &= \frac{2}{3} \\[4pt] A'B = BC' = A'C' &= \frac{2}{3} \times 5 \\[4pt] &= \frac{10}{3} \\[4pt] &\approx 3.33 \end{aligned} \\[4pt] \textbf{Answer:} & \text{The required triangle } A'BC' \text{ is equilateral with each side } \dfrac{10}{3} \text{ cm} \approx 3.33 \text{ cm.} \end{array} \]