The ratio of the corresponding altitudes of two similar triangles is \(\dfrac{3}{5}\). Is it correct to say that ratio of their areas is \(\dfrac{6}{5}\)? Why?
No. The area ratio is \(\left(\dfrac{3}{5}\right)^2=\dfrac{9}{25}\).
Step 1: In similar triangles, the ratio of any two corresponding linear dimensions (like sides, altitudes, medians, etc.) is the same. Here, the ratio of altitudes is given as \(\dfrac{3}{5}\).
Step 2: When we compare areas of two similar triangles, the ratio of areas is equal to the square of the ratio of their corresponding sides (or altitudes).
Step 3: So, the ratio of areas = \(\left(\dfrac{3}{5}\right)^2\).
Step 4: Calculate the square: \(\left(\dfrac{3}{5}\right)^2 = \dfrac{3^2}{5^2} = \dfrac{9}{25}\).
Step 5: The student’s guess \(\dfrac{6}{5}\) is wrong because areas do not compare in the same ratio as altitudes. They always compare by the square of that ratio.
Final Result: The ratio of their areas is \(\dfrac{9}{25}\), not \(\dfrac{6}{5}\).