In parallelogram ABCD, the diagonals intersect at O. Given AC = 10, BD = 18, and AD = 12, find the perimeter of triangle BOC.

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 $CO = \frac{1}{2} \times 10 = 5$ and $BO = \frac{1}{2} \times 18 = 9$.

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 $9 + 5 + 12 = 26$.

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