In △ABC, AD is the bisector of ∠A, meeting side BC at D. If AB=10 cm, AC=14 cm, and BC=6 cm, find the lengths of segments BD and DC.
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Step-by-Step Solution
Step 1: Apply the Angle Bisector Theorem
The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. In △ABC, AD bisects ∠A, so it divides side BC into segments BD and DC such that their ratio is equal to the ratio of the other two sides, AB and AC.
Step 2: Substitute given values
We are given the lengths of sides AB=10 cm and AC=14 cm. Substitute these values into the proportion derived from the Angle Bisector Theorem.
Step 3: Simplify the ratio and express BD in terms of DC
Simplify the ratio 1410 to 75. Then, rearrange the equation to express BD in terms of DC. This will be useful for solving the system of equations.
Step 4: Use the total length of BC
We know that point D lies on the side BC, so the sum of the lengths of segments BD and DC must be equal to the total length of BC. We are given BC=6 cm.
Step 5: Solve for DC
Substitute the expression for BD from step 3 into the equation from step 4. Combine the terms involving DC and solve for DC.
Step 6: Solve for BD
Now that we have the value of DC, we can use the equation BC=BD+DC to find the length of BD.