1A right-angled triangle ABC has the right angle at B, AB = 4 cm and BC = 12 cm. Find its area.
2Triangle ABC has base BC = 7 cm and the perpendicular AD from A to BC is 4 cm. Find its area.
3PQR is an equilateral triangle with QR = 12 cm. Find its area.
4Trapezium ABCD has AD // BC, AD = 6 cm, BC = 9 cm and height AE = 5 cm. Find its area.
5Parallelogram PQRS has base QR = 6 in. and height PM = 7 in. Find its area.
6EFGH is a parallelogram with FG = 6 cm, EF = 5 cm and diagonal EG = EF. Find its area.
7WXYZ is a square with diagonal WY = \( 2\sqrt{2} \) cm. PXYW is a parallelogram with P on the line ZW produced. Find the area of parallelogram PXYW.
8In parallelogram ABCD, E is on AD, BE = EC = 10 cm, EF is perpendicular to BC and AD = 16 cm. Find the area of ABCD.
9PQST is a parallelogram and R is on TS produced. SA is perpendicular to TP, SA = 8 cm and the area of \( \triangle PQR \) is 64 cm². Find TP.
10PQRS is a parallelogram and O is any point on QR. PO is produced to meet SR produced at T, and Q and T are joined. Prove that area of \( \triangle QOT \) = area of \( \triangle ROS \).
11B, C and E lie on one line with BC = EC, and AC // DE. If the area of \( \triangle ACE \) is 24 cm², find the area of quadrilateral ABCD.
12In trapezium PQRS, PS // QR, PQ = 13 cm, PS = 18 cm, QR = 23 cm and SR is perpendicular to QR. Find the area of \( \triangle PSR \).
13PQ // SR and PS // QT, with T on SR. The area of trapezium PQRS is 95 cm² and the area of \( \triangle QRT \) is 15 cm². Find the area of \( \triangle RPQ \).
14ABCD is a parallelogram, E is on DC produced so that DE // AB, and F is the midpoint of BE. If the area of \( \triangle AFE \) is 12 cm², find the area of parallelogram ABCD.
15In parallelogram ABCD, E is a point on AD with angle BEC = 90°, BE = 6 cm and CE = 9 cm. Find the area of ABCD.
16PQRS is a parallelogram and QRXY is a square, with P, Y, S and X on one line. Prove that the area of parallelogram PQRS equals the area of square QRXY.
17ABCD is a quadrilateral. B, C and E lie on one line and DE is drawn so that AC // DE. Prove that area of quadrilateral ABCD = area of \( \triangle ABE \).
18ABCD and EFGC are parallelograms, with E on AD and B on FG. Prove that area of parallelogram ABCD = area of parallelogram EFGC.
19PQRS is a parallelogram and SR is extended to T. P and T are joined, and PT meets QR at O. Prove that area of \( \triangle POR \) = area of \( \triangle QOT \).
20PQRS is a parallelogram in which SR is extended to T, and PT and ST are joined, with PT meeting QR at O. Prove that area of \( \triangle QOT \) = area of \( \triangle ROS \).
21ABCD is a parallelogram, X is a point on AD and Y is a point on DC. Triangles XBC and YAB are drawn. Prove that area of \( \triangle ABY \) = area of \( \triangle ABX \) + area of \( \triangle CDX \).
22ABCD is a parallelogram and E is any point on BC. DE and AB are produced to meet at F, and C and F are joined. Prove that area of \( \triangle CEF \) = area of \( \triangle ABE \).
23ABCD is a trapezium with AD // PQ // BC, where P is on AB and Q is on DC. Prove that area of \( \triangle AQB \) = area of \( \triangle DPC \).
24In a figure QT // RS, PR // TS and PS // QR. Prove that area of \( \triangle PQR \) = area of \( \triangle PTS \).
25ABCD is a trapezium in which Y is a point on AC and X is a point on BD, with AD // XY // BC. Prove that area of \( \triangle AXC \) = area of \( \triangle BYD \).
26ABCD is a quadrilateral in which DB // CE, with E on AB produced. Prove that the area of \( \triangle ADE \) equals the area of quadrilateral ABCD.
27ABCDEF is a hexagon in which AF // BE // CD and AB // CF // DE, and BE meets CF at P. If the areas of parallelograms ABPF and CDEP are equal, prove that EF // BC.
28In quadrilateral PQRS the diagonal QS bisects the other diagonal PR at M. Prove that the area of \( \triangle PQS \) is half the area of quadrilateral PQRS.
29In triangle ABC, P is the midpoint of AB, Q is any point on BC, R is a point on AB between P and A, and PQ // RC. Prove that area of \( \triangle BQR = \frac{1}{2} \) area of \( \triangle ABC \).
30ABCD is a trapezium with AD // BC. E is a point on AD and F is a point on DC. If the areas of \( \triangle ABE \) and \( \triangle ACF \) are equal, prove that EF // AC.
31Two triangles stand on the same base between the same parallels. Must they be congruent? Explain.
32Two parallel lines are 7 cm apart. A triangle stands on a base of 10 cm on one line with its apex on the other. A second triangle stands on the same base with its apex elsewhere on the same line. Find the area of each.
33In the proof of Theorem 3, why is the construction AD // BC needed?
34How do you decide whether to add or to subtract a common triangle in an area proof?