In worksheet on area of polygon we will find the area of triangle, quadrilateral, pentagon etc,. using the formula of area of the triangle formed by three coordinate points.
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1. A (1, 5), B (3, 1) and C (5, 7) are the vertices of the ∆ ABC. If D, E and F are the midpoints of the sides BC, CA and AD respectively, find the area of the ∆ DEF. Show also that, ∆ ABC = 4 ∆ DEF.
2. The coordinates of A, B, C are (6, 3), (3, 5) and (4, 2) respectively and P is the point (x, y) ; show that,
(area of the ∆ PBC)/(area of the ∆ ABC) = (x + y  2)/7
3. The vertices A, B and C of the ∆ABC have coordinates (3, 2,), (2, 2) and (6, 1) respectively. Find the area of the ∆ ABC and the length of the perpendicular from A on BC.
4. It the points A and B have coordinates (a cos θ, b sin θ) and (a sinθ, b cos θ) respectively and O is the origin, then show that the area of the ∆ OAB is independent of θ.
5. The points P, Q, R are collinear; if the coordinates of P and Q be (3, 4)and (7, 7) respectively and PR = 10 units, find the coordinates of R.
6. The coordinates of the points A, B, C, D are respectively (6, 3), (3, 5), (4, 2) and (x, 3x); if (area of the ∆ PBC)/(area of the ∆ ABC)= 1/2, find x.
7. The coordinates of the points A, B, C and D are (2, 3), (8, 9), (0, 4) and (3, 0) respectively. Find the ratio in which the linesegment AB is divided by the segment, CD.
8. The coordinates of the points A and B are (3, 4) and (5, 2) respectively; if PA = PB and the area of the ∆ PAR = 10 sq. units, find the coordinates of P.
9. Find the area of the quadrilateral whose vertices have coordinates:
(i) (1, 1), (3, 4), (5, 2) and (1, 7).
(ii) (1, 4), (2, 1), (2, 3) and (3, 3).
10. The coordinates of the vertices A, B, C and D of the quadrilateral ABCD are (1, 2), (5, 6),
(7, 4) and (k, 2); if the area of the quadrilateral be zero, then find the value of k.
11. The area of a quadrilateral is 28 sq. units. If the coordinates of its angular points be (1, 6), (2, 4), (3, 2) and (a, b), then show that, 2a + b = 6 or, 2a + b + 22 = 0.
12. Show that the area of the quadrilateral whose vertices taken in order are (a, 0), (b, 0), (0, a) and (0, b) is zero (a > 0, b> 0). Give the geometrical significance of the result.
13. Find the area of the pentagon whose vertices have coordinates
(0, 1), (2, 3), (5, 4), (4, 0) and (3, 2).
Answers for the worksheet on area of polygon are given below to check the exact answers of the above questions.
1. 4 sq units
3. 7.5 sq. units, 3 units
5. (11, 10) or, ( 5,  2)
6. 11/8
7. 11 : 47
8. (7, 2) or , (1, 0) ;
9. (i) 20.5 sq. units
(ii) 18.5 sq. units ;
10. K = 3.
13. 16 sq. units
● Coordinate Geometry
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