plot of land ABCD such that AB = 85m, BC = 75m, CD = 60m, DA = 50m and angle ACB = . Determine the area of the plot in hectares correct to two decimal places
To determine the area of the plot of land, we can use the formula for the area of a quadrilateral if we have the lengths of its sides and one of its angles.
Step 1: Draw the diagram.
Start by drawing a diagram of the plot of land ABCD. Ensure that the lengths of the sides AB, BC, CD, and DA are correctly measured.
Step 2: Determine the height of the quadrilateral.
To find the height of the quadrilateral, draw a perpendicular line from point C to line AD. Let's call the intersection point E.
Step 3: Find the length of CE.
Since angle ACB is not given, we need to find it first. To do that, we can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C),
where c is the side opposite the angle C, and a and b are the other two sides.
In this case, side AB = 85m, side BC = 75m, and side AC (which is unknown) is opposite angle ACB.
Let's solve for angle ACB:
75^2 = 85^2 + 60^2 - 2 * 85 * 60 * cos(C)
5625 = 7225 + 3600 - 10200cos(C)
10200cos(C) = 7225 + 3600 - 5625
10200cos(C) = 5200
cos(C) = 5200 / 10200 = 0.5098
C ≈ arccos(0.5098) ≈ 59.60°
Now that we have angle ACB, we can use trigonometry to find CE:
sin(C) = CE / BC
sin(59.60°) = CE / 75
CE = 75 * sin(59.60°) ≈ 64.95m
Step 4: Calculate the area of the quadrilateral.
The area of a quadrilateral can be determined by dividing it into two triangles and finding the sum of their areas.
Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle CDA
Area of triangle ABC = (1/2) * AB * CE
Area of triangle CDA = (1/2) * CD * CE
Let's calculate the area:
Area of triangle ABC = (1/2) * 85 * 64.95 ≈ 2773.63m^2
Area of triangle CDA = (1/2) * 60 * 64.95 ≈ 1946.50m^2
Area of quadrilateral ABCD = 2773.63 + 1946.50 ≈ 4719.13m^2
Step 5: Convert the area to hectares.
Since we want the area in hectares, we need to divide by 10,000 (since 1 hectare = 10,000 square meters):
Area in hectares = 4719.13 / 10,000 ≈ 0.47 hectares
Therefore, the area of the plot of land ABCD is approximately 0.47 hectares, correct to two decimal places.