NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 10: Construction - Exercise 10.3Question 3
Class 10 - Mathematics - CHAPTER 10: Construction - Exercise 10.3
Question. 3
Draw a triangle \(ABC\) in which \(BC=6\,\text{cm},\; CA=5\,\text{cm}\) and \(AB=4\,\text{cm}.\) Construct a triangle similar to it with scale factor \(\dfrac{5}{3}\).
Answer:
- Construct \(\triangle ABC\) with the given side lengths.
- From vertex \(A\), draw a ray \(AX\) making an acute angle with \(AB\).
- On \(AX\), mark five equal points \(A_1,\ldots,A_5\).
- Join \(A_5\) to \(B\) and \(C\). Through \(A_3\) draw \(A_3B'\,\parallel\,A_5B\) and \(A_3C'\,\parallel\,A_5C\).
- Then \(\triangle AB'C'\sim\triangle ABC\) with scale factor \(\dfrac{5}{3}\) (enlargement).
Detailed Answer with Explanation:
Step 1: We are asked to enlarge triangle \(ABC\) in the ratio \(5:3\). That means each side of the new triangle must be \(\dfrac{5}{3}\) times the side of the original triangle.
Step 2: To do this, we use the method of dividing a line into equal parts with the help of a ray and parallel lines.
Step 3: On the ray \(AX\), we marked 5 equal parts because the numerator of the ratio is 5.
Step 4: We join the 5th point (\(A_5\)) with \(B\) and \(C\). This gives the full size corresponding to '5' in the ratio.
Step 5: Since the ratio is \(\tfrac{5}{3}\), we take the 3rd division point (\(A_3\)) to represent the smaller triangle. By drawing parallels from \(A_3\), we ensure that the sides are reduced proportionally.
Step 6: The triangle \(AB'C'\) formed in this way is similar to the original triangle \(ABC\), and the sides are enlarged in the ratio \(\tfrac{5}{3}\).
Reason: This works because of the Basic Proportionality Theorem. Drawing parallels creates smaller triangles that are similar to the bigger triangle, keeping the ratio of corresponding sides exactly as required.