In △ABC, a line DE is drawn parallel to the base BC, intersecting side AB at point D and side AC at point E.If AD=x, DB=x−2, AE=x+2, and EC=x−1, find the value of x and the total length of side AB.
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Step-by-Step Solution
Step 1: Apply Basic Proportionality Theorem (BPT)
Since the line segment DE is parallel to the base BC in △ABC, we can apply the Basic Proportionality Theorem (also known as Thales Theorem). This theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Step 2: Substitute the given values
Now, we substitute the given expressions for the lengths of the segments into the proportionality equation. We have AD=x, DB=x−2, AE=x+2, and EC=x−1.
Step 3: Solve the equation for x
To solve for x, we cross-multiply the terms in the equation. This leads to a quadratic equation, which simplifies to a linear equation after cancelling out the x2 terms. We then isolate x to find its value.
Step 4: Calculate the lengths of the segments
Now that we have found the value of x=4, we can substitute this value back into the expressions for the lengths of the segments to find their numerical values. This also helps verify that all lengths are positive, as required for geometric segments.
Step 5: Calculate the total length of side AB
The total length of side AB is the sum of the lengths of its segments AD and DB. We add the calculated numerical values to find the total length.