In a rectangle ABCD, E is the midpoint of AD and F is the midpoint of CE. If the area of triangle BDF is 24 sq. cm, what is the area (in sq. cm) of rectangle ABCD?

Step 1: Understand the problem We have a rectangle ABCD. E is the midpoint of AD and F is the midpoint of CE. The area of triangle BDF is given as 24 sq. cm. We need to find the area of the rectangle ABCD.

Step 2: Visualize the problem Draw the rectangle ABCD. Mark E as the midpoint of AD and F as the midpoint of CE. Draw the triangle BDF.

Step 3: Analyze the problem Since E is the midpoint of AD, DE = AE = 1/2 AD. Similarly, since F is the midpoint of CE, CF = FE = 1/2 CE. Therefore, triangle BDF is similar to triangle BAC by AA similarity (Angle-Angle similarity) as they share an angle at B and angle ABD is equal to angle DBF (alternate angles for parallel lines AB and DF with transversal BD).

Step 4: Use the properties of similar triangles The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the area of triangle BAC is four times the area of triangle BDF (since the ratio of their sides is 2:1, the square of this ratio is 4:1).

Step 5: Calculate the area of triangle BAC Since the area of triangle BDF is 24 sq. cm, the area of triangle BAC is 4 * 24 = 96 sq. cm.

Step 6: Use the properties of rectangles The area of a rectangle is equal to the sum of the areas of the two triangles formed by drawing a diagonal. Therefore, the area of rectangle ABCD is 2 * the area of triangle BAC.

Step 7: Calculate the area of rectangle ABCD The area of rectangle ABCD is 2 * 96 = 192 sq. cm.