Draw a right triangle \(ABC\) in which \(BC=12\,\text{cm},\; AB=5\,\text{cm}\) and \(\angle B=90^{\circ}\). Construct a triangle similar to it with scale factor \(\dfrac{2}{3}\). Is the new triangle also a right triangle?
Yes. Similarity preserves angles, hence the image of \(\angle B=90^{\circ}\) is also \(90^{\circ}\); the new triangle is right-angled at \(B\).
Step 1: A scale factor of \(\tfrac{2}{3}\) means every side in the new triangle is reduced to \(\tfrac{2}{3}\) of the original length.
Step 2: We use the method of dividing a ray into equal parts. Here the denominator (3) tells us to mark 3 equal parts, and the numerator (2) tells us to stop at the 2nd division.
Step 3: By joining and drawing parallels, the sides shrink in the correct ratio. This ensures the triangles are similar.
Step 4: In similar triangles, all angles remain the same. Since \(\angle B\) was \(90^{\circ}\) in the original triangle, it remains \(90^{\circ}\) in the new triangle.
Therefore, the constructed triangle is also a right triangle.