In △ ABC, D is a point on side BC such that
DCBD=23.
If the area of △ ABD is 54\ cm2 answer the following:
1. Find the area of △ ACD.
2. Find the area of △ ABC.
3. A point E lies on AB such that DE ∥AC. Find the ratio BE : BA.
4. Hence, find the ratio of the areas of △ BDE and △ BAC.
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Step-by-Step Solution
Step 1: Relate Areas of Triangles with Same Height
When two triangles share the same vertex and their bases lie on the same straight line, their heights are equal. In this case, triangles ABD and ACD share vertex A and their bases BD and DC lie on the line BC. Therefore, the ratio of their areas is equal to the ratio of their bases.
Step 2: Calculate Area of Triangle ACD
We are given that the area of △ABD is 54 cm2 and the ratio BD:DC=3:2. By substituting these values into the area ratio formula, we can calculate the area of △ACD.
Step 3: Calculate Area of Triangle ABC
The area of △ABC is the sum of the areas of △ABD and △ACD, as point D lies on side BC.
Step 4: Find Ratio BE : BA using Thales Theorem
Since DE∥AC, by the Basic Proportionality Theorem (also known as Thales Theorem), the line DE divides the sides AB and BC in the same ratio. This means BE/EA=BD/DC. From this, we can find the ratio BE:BA.
Step 5: Find Ratio of Areas of Triangles BDE and BAC
Since DE∥AC, △BDE is similar to △BAC. For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. We found BE:BA=3:5, so we can use this ratio.