Area Of Triangles And Quadrilaterals (Shoelace Formulae)

1. Given that A = (3, 4), B = (-4, 2), C= (6, -1) and D = (20, 3) , find the area of quadrilateral ABCD. Area of quadrilateral ABCD = ______ sq. units.

To find the area of a quadrilateral, we can use the Shoelace Formula. The formula states that the area of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), and (x4, y4) is equal to 1/2 * |(x1 * y2 + x2 * y3 + x3 * y4 + x4 * y1) - (y1 * x2 + y2 * x3 + y3 * x4 + y4 * x1)|. Using the given coordinates, we can substitute the values into the formula and calculate the area. In this case, the area of quadrilateral ABCD is 61.5 square units.

Explanation

To find the area of a quadrilateral, we can use the Shoelace Formula. The formula states that the area of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), and (x4, y4) is equal to 1/2 * |(x1 * y2 + x2 * y3 + x3 * y4 + x4 * y1) - (y1 * x2 + y2 * x3 + y3 * x4 + y4 * x1)|. Using the given coordinates, we can substitute the values into the formula and calculate the area. In this case, the area of quadrilateral ABCD is 61.5 square units.

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3. A triangle ABC with the given vertices A= (1, 3), B =(5, 1) and C = (3, r) in an anticlocwise direction. Find the value of r when its area is 6 square units . Hence , r = ____.

In order to find the value of r, we need to calculate the area of the triangle using the given coordinates of the vertices. The formula for the area of a triangle is 1/2 * base * height. The base can be found by calculating the distance between points A and B, which is 4 units. The height can be found by calculating the distance between point C and the line AB, which is the perpendicular distance from point C to line AB. By using the formula for the distance between a point and a line, we can find that the height is 2 units. Therefore, the area of the triangle is 1/2 * 4 * 2 = 4 square units. Since the given area is 6 square units, it is not possible to find a value of r that satisfies this condition. Therefore, the answer is not available.

Explanation

In order to find the value of r, we need to calculate the area of the triangle using the given coordinates of the vertices. The formula for the area of a triangle is 1/2 * base * height. The base can be found by calculating the distance between points A and B, which is 4 units. The height can be found by calculating the distance between point C and the line AB, which is the perpendicular distance from point C to line AB. By using the formula for the distance between a point and a line, we can find that the height is 2 units. Therefore, the area of the triangle is 1/2 * 4 * 2 = 4 square units. Since the given area is 6 square units, it is not possible to find a value of r that satisfies this condition. Therefore, the answer is not available.

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4. Find the area of triangle ABC, given that A = (2, -3) , B = (-2, -1) and C = (4, 3). Hence, the area of triangle ABC = ________ sq. units

To find the area of a triangle, we can use the formula: Area = 1/2 * base * height. In this case, we can take any two sides of the triangle as the base and height. Let's take AB as the base and the distance between AB and point C as the height. The distance between two points can be found using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Using the distance formula, we find that the distance between AB and C is sqrt((4 - 2)^2 + (3 - (-3))^2) = sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 2sqrt(10).

Now, we can substitute the values into the area formula: Area = 1/2 * AB * height = 1/2 * sqrt((2 - (-2))^2 + (-3 - (-1))^2) * 2sqrt(10) = 1/2 * sqrt(4^2 + 2^2) * 2sqrt(10) = 1/2 * sqrt(16 + 4) * 2sqrt(10) = 1/2 * sqrt(20) * 2sqrt(10) = sqrt(20) * sqrt(10) = 2sqrt(2) * sqrt(10) = 2 * sqrt(2 * 10) = 2 * sqrt(20) = 2 * 2sqrt(5) = 4sqrt(5).

Therefore, the area of triangle ABC is 4sqrt(5) square units. However, the given answer is 14, which is incorrect.

Explanation

To find the area of a triangle, we can use the formula: Area = 1/2 * base * height. In this case, we can take any two sides of the triangle as the base and height. Let's take AB as the base and the distance between AB and point C as the height. The distance between two points can be found using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Using the distance formula, we find that the distance between AB and C is sqrt((4 - 2)^2 + (3 - (-3))^2) = sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 2sqrt(10).

Now, we can substitute the values into the area formula: Area = 1/2 * AB * height = 1/2 * sqrt((2 - (-2))^2 + (-3 - (-1))^2) * 2sqrt(10) = 1/2 * sqrt(4^2 + 2^2) * 2sqrt(10) = 1/2 * sqrt(16 + 4) * 2sqrt(10) = 1/2 * sqrt(20) * 2sqrt(10) = sqrt(20) * sqrt(10) = 2sqrt(2) * sqrt(10) = 2 * sqrt(2 * 10) = 2 * sqrt(20) = 2 * 2sqrt(5) = 4sqrt(5).

Therefore, the area of triangle ABC is 4sqrt(5) square units. However, the given answer is 14, which is incorrect.

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5. Given that A = (2, -3), B = (3, -1) , C = (2, 0) and D = (-1, 1) . Find the area of triangle ABC : the area of triangle ACD. Area of triangle ABC : Area of triangle ACD = ____ : ____

A. 1 : 2

B. 2 : 1

C. 1 : 3

D. None of the above

The correct answer is C. 1 : 3.

To find the area of triangle ABC, we can use the formula for the area of a triangle given three vertices. The coordinates of A, B, and C are given as (2, -3), (3, -1), and (2, 0) respectively. Using these coordinates, we can calculate the length of the base AB and the height from C to the line AB. Then, we can use the formula for the area of a triangle (1/2 * base * height) to find the area of triangle ABC.

Similarly, to find the area of triangle ACD, we can use the coordinates of A, C, and D which are (2, -3), (2, 0), and (-1, 1) respectively. By calculating the length of the base AC and the height from D to the line AC, we can find the area of triangle ACD using the same formula.

Comparing the two areas, we can see that the area of triangle ABC is 1/3 of the area of triangle ACD. Therefore, the ratio of the area of triangle ABC to the area of triangle ACD is 1 : 3.

Explanation

The correct answer is C. 1 : 3.

To find the area of triangle ABC, we can use the formula for the area of a triangle given three vertices. The coordinates of A, B, and C are given as (2, -3), (3, -1), and (2, 0) respectively. Using these coordinates, we can calculate the length of the base AB and the height from C to the line AB. Then, we can use the formula for the area of a triangle (1/2 * base * height) to find the area of triangle ABC.

Similarly, to find the area of triangle ACD, we can use the coordinates of A, C, and D which are (2, -3), (2, 0), and (-1, 1) respectively. By calculating the length of the base AC and the height from D to the line AC, we can find the area of triangle ACD using the same formula.

Comparing the two areas, we can see that the area of triangle ABC is 1/3 of the area of triangle ACD. Therefore, the ratio of the area of triangle ABC to the area of triangle ACD is 1 : 3.

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