Question

Construct \( \triangle ABC \) in which \( BC = 6 \, \text{cm} \), \( AB = 3 \, \text{cm} \), and \( \angle ABC = 45^\circ \). Construct a similar triangle \( A'B'C' \) whose corresponding sides are \( \frac{3}{4} \) of the sides of \( \triangle ABC \). Write the steps of construction in brief.

Hint:

When constructing similar triangles, ensure that the corresponding angles are equal, and the corresponding sides are proportional.

Answer 1

Step 1: Draw the base of the triangle.
Draw a line segment \( BC = 6 \, \text{cm} \). This will be the base of \( \triangle ABC \).
Step 2: Draw \( \angle ABC = 45^\circ \).
At point \( B \), use a protractor to measure an angle of \( 45^\circ \). Draw a line from \( B \) at this angle.
Step 3: Mark the point \( A \).
Using a compass, place the pointer at point \( B \) and set the radius to 3 cm. Draw an arc along the line. Mark the point where the arc intersects the line as point \( A \).
Step 4: Join points to form \( \triangle ABC \).
Now, join the points \( A \) and \( C \) to complete \( \triangle ABC \).
Step 5: Construct a similar triangle.
To construct a triangle similar to \( \triangle ABC \) whose corresponding sides are \( \frac{3}{4} \) of the sides of \( \triangle ABC \), we need to reduce each side by the ratio \( \frac{3}{4} \).
Step 6: Draw a line parallel to \( BC \).
Draw a line parallel to \( BC \) passing through \( A \). The distance between this new line and the original line \( BC \) will be \( \frac{3}{4} \) of the distance between the original triangle's corresponding vertices.
Step 7: Mark corresponding points.
Measure and mark the points on the parallel line such that the sides of the new triangle \( A'B'C' \) are \( \frac{3}{4} \) of the original triangle's sides.
Conclusion:
The similar triangle \( A'B'C' \) is now constructed, with corresponding sides that are \( \frac{3}{4} \) of the sides of \( \triangle ABC \).