In parallelogram ABCD, the diagonals intersect at O. Given AC = 10, BD = 18, and AD = 12, find the perimeter of triangle BOC.
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Step-by-Step Solution
Step 1: Identify properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the point where the diagonals intersect (point O) divides each diagonal into two equal halves.
Step 2: Calculate lengths of BO and CO
Since O is the midpoint of AC and BD, we can find the lengths of CO and BO. Given AC=10 and BD=18, we have CO=21×10=5 and BO=21×18=9.
Step 3: Identify side BC
In a parallelogram, opposite sides are equal in length. Therefore, the length of side BC is equal to the length of side AD. Given AD=12, we have BC=12.
Step 4: Calculate the perimeter of triangle BOC
The perimeter of a triangle is the sum of the lengths of its three sides. For triangle BOC, the perimeter is the sum of BO, CO, and BC. Substituting the calculated values, we get 9+5+12=26.