In triangle PQR, M, N, and S are the mid-points of sides PQ, PR, and QR, respectively. If the area of triangle PQR is 46 cm², what is the area (in cm²) of the quadrilateral MNRQ?

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Given triangle PQR with area 46 cm². Points M, N, and S are midpoints of sides PQ, PR, and QR respectively.

Step 1: Since M and N are midpoints of PQ and PR, segment MN is parallel to QR and MN = 1/2 * QR (Midpoint theorem).

Step 2: Quadrilateral MNRQ is formed by points M, N, R, and Q.

Step 3: Consider triangle PQR. The triangle MNS formed by midpoints M, N, and S is similar to triangle PQR and its area is 1/4 of the area of PQR because each side is half the length of the corresponding side in PQR.

Step 4: Area of triangle MNS = (1/4) * 46 = 11.5 cm².

Step 5: Quadrilateral MNRQ can be seen as the area of triangle PQR minus the area of triangle MNS.

Step 6: Area of quadrilateral MNRQ = 46 - 11.5 = 34.5 cm².

However, this contradicts the options given. Let's analyze carefully.

Step 7: Points M and N are midpoints of PQ and PR, so MN is parallel to QR and MN = 1/2 QR.

Step 8: Quadrilateral MNRQ consists of triangle MNR and triangle NRQ.

Step 9: Triangle MNR is similar to triangle PQR but scaled down by 1/2 in two dimensions, so area is 1/4 of PQR.

Step 10: Triangle NRQ is half of triangle PRQ because N is midpoint of PR.

Step 11: Area of triangle MNR = 1/4 * 46 = 11.5 cm².

Step 12: Area of triangle NRQ = 1/2 * area of triangle PRQ = 1/2 * (area of PQR) = 1/2 * 46 = 23 cm².

Step 13: Quadrilateral MNRQ = area of triangle MNR + area of triangle NRQ = 11.5 + 23 = 34.5 cm².

Step 14: But this is not consistent with the shape. Actually, point S is midpoint of QR, so quadrilateral MNRQ includes points M, N, R, and Q.

Step 15: Quadrilateral MNRQ = triangle PQR - triangle MNS.

Step 16: Area of triangle MNS = 1/4 * 46 = 11.5 cm².

Step 17: Therefore, area of quadrilateral MNRQ = 46 - 11.5 = 34.5 cm².

Step 18: The correct answer is 34.5 cm².

Hence, option 4 is correct.

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