Draw a line segment of length 7 cm. Find a point \(P\) on it which divides it in the ratio \(3:5\).
Check (optional): \(AP = \dfrac{3}{8}\times 7 = 2.625\,\text{cm},\; BP = \dfrac{5}{8}\times 7 = 4.375\,\text{cm}.\)
Step 1: We need to divide the line \(AB\) in the ratio \(3:5\). The total of the ratio is \(3+5 = 8\).
Step 2: By drawing 8 equal steps on the ray from \(A\), each step represents 1 part of the ratio.
Step 3: The point \(A_3\) corresponds to 3 parts from \(A\). When we join \(A_8\) to \(B\) and draw a parallel line from \(A_3\), the parallel line ensures similar triangles are formed.
Step 4: In similar triangles, the sides are in proportion. This makes the division on \(AB\) exactly the same as the ratio of steps (3 parts to 5 parts).
Step 5: Therefore, point \(P\) divides the line as required: \(AP : PB = 3 : 5\).
Why this works: The method of marking equal steps and drawing parallels is based on the concept of similar triangles. Parallel lines create equal ratios of sides, so the internal division comes out correctly.