In Δ ABC, line DE∥BC. If AD = 3x - 2, AE = 5x - 4, BD = 7x - 5, and CE = 5x - 3, find the value of x using the Basic Proportionality Theorem (BPT). The solution yields x=1 or x=107.
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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=3x−2, BD=7x−5, AE=5x−4, and CE=5x−3.
Step 3: Solve the equation for x
To solve for x, we cross-multiply the terms in the equation. Then, we expand both sides and rearrange the terms to form a quadratic equation in the standard form ax2+bx+c=0.
Step 4: Factor the quadratic equation
We solve the quadratic equation 10x2−17x+7=0 by factoring. We look for two numbers that multiply to 10×7=70 and add up to −17. These numbers are −10 and −7. We then split the middle term and factor by grouping.
Step 5: Find the possible values of x
From the factored form (10x−7)(x−1)=0, we set each factor equal to zero to find the possible values of x. This gives us two potential solutions for x.
Step 6: Check for valid lengths
For the lengths of the segments to be valid, they must be positive. We test both values of x. When x=107, BD becomes negative, which is not possible for a length. Therefore, x=107 is not a valid solution. When x=1, all segment lengths are positive, making it a valid solution.