In △ABC, AD is the bisector of ∠A, meeting side BC at D. If AB=12 cm, AC=18 cm, and BC=10 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, since AD bisects ∠A, the ratio of BD to DC is equal to the ratio of AB to AC.
Step 2: Substitute given values
We are given the lengths of sides AB=12 cm and AC=18 cm. We substitute these values into the proportion derived from the Angle Bisector Theorem.
Step 3: Simplify the ratio
We simplify the ratio 1812 by dividing both the numerator and the denominator by their greatest common divisor, which is 6. This gives us a simplified ratio of 32.
Step 4: Express BD in terms of DC
From the simplified ratio DCBD=32, we can express BD in terms of DC by multiplying both sides by DC. This relationship will be used in the next step.
Step 5: Use the total length of BC
We know that point D lies on 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=10 cm. We substitute BD=32DC into the equation BD+DC=10 and solve for DC.
Step 6: Solve for DC and BD
To find DC, we multiply both sides of the equation 35DC=10 by 53. Once DC is found, we substitute its value back into the equation BD=32DC to find BD.