Question : PQRA is a rectangle, AP = 22 cm and PQ = 8 cm. $\triangle$ABC is a triangle whose vertices lie on the sides of PQRA such that BQ = 2 cm and QC = 16 cm. Then the length of the line joining the midpoints of the sides AB and BC is:
Option 1: $4\sqrt2$ cm
Option 2: $5$ cm
Option 3: $6$ cm
Option 4: $10$ cm
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Correct Answer: $5$ cm
Solution : Given: PQRA is a rectangle, AP = 22 cm and PQ = 8 cm. $\triangle$ABC is a triangle whose vertices lie on the sides of PQRA such that BQ = 2 cm and QC = 16 cm. Let D and E be the two midpoints of BC and AB, respectively. PQRA is a rectangle, PA = QR = 22 cm, PQ = AR = 8 cm and QC = 16 cm. So, CR = (22 – 16) = 6 cm From $\triangle$ACR, AC = $\sqrt{8^2+6^2}$ ⇒ AC = 10 cm Since D and E are the two midpoints of BC and AB, respectively, So, DE = $\frac{AC}{2}$ ⇒ DE = $\frac{10}{2}$ ⇒ DE = 5 cm Hence, the correct answer is 5 cm.
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