Question

Find the area of quadrilateral formed by joining the points \((-4,2),(1,-1),(4,1)\) and \((2,5):\) (a) \(25.4\) (b) \(20.5\) (c) \(24.5\) (d) none of these

Short answer

In a quadrilateral with vertices A(-4,2), B(1,-1), C(4,1), and D(2,5), the area of the quadrilateral can be found by first determining its type, calculating the side lengths, and then using the appropriate formula. In this case, the quadrilateral is a parallelogram with an area of approximately 21.2, which is not among the given options. Hence, the correct answer is (d) none of these.

Step-by-step solution

Calculate the lengths of the sides

We are given the coordinates of the four vertices of the quadrilateral: \(A(-4,2)\), \(B(1,-1)\), \(C(4,1)\), and \(D(2,5)\). We will use the distance formula to find the lengths. $$ AB = \sqrt{(1-(-4))^2+(-1-2)^2} = 5\sqrt{2} $$ $$ BC = \sqrt{(4-1)^2+(1-(-1))^2} = 3\sqrt{2} $$ $$ CD = \sqrt{(2-4)^2+(5-1)^2} = 2\sqrt{5} $$ $$ DA = \sqrt{(-4-2)^2+(2-5)^2} = 3\sqrt{5} $$

Determine the type of quadrilateral

Since the lengths of the opposite sides are equal (\(AB = CD = 5\sqrt{2}\) and \(DA = BC = 3\sqrt{2}\)), this is a parallelogram.

Calculate the area of the parallelogram

To find the area of the parallelogram, we need to find the base length and the height. We can consider \(AB\) as the base and calculate the length of the altitude from \(C\) to the side \(AB\). Let the foot of the altitude be \(E\). To find the coordinates of \(E\), we need to find the equation of the line \(AB\). To do this, we first find the slope of the line \(AB\). $$ m_{AB} = \frac{-1-2}{1-(-4)} = \frac{-3}{5} $$ The equation of line \(AB\) will be in the form: $$ y - y_1 = m(x - x_1) $$ Plugging the coordinate of point \(A(-4,2)\) and the slope \(m_{AB} = -\frac{3}{5}\), we get: $$ y - 2 = -\frac{3}{5}(x + 4) $$ $$ 5(y - 2) = -3(x + 4) $$ Now calculating the distance between the point \(C(4,1)\) and line \(AB\) (the length of \(CE\)) using the formula: $$ d = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}} $$ We plug \(a = -3\), \(b = 5\), \(c = 6\), \(x = 4\), and \(y = 1\) into the formula and get the length of \(CE\): $$ d = 3 = CE $$ Now that we have the altitude length \(CE\) and the base \(AB\), we can calculate the area of the parallelogram as: $$ Area = base * height = 5\sqrt{2} * 3 = 15\sqrt{2} $$ which is approximately \(21.2\). The correct answer is (d) none of these, since the area of the quadrilateral is approximately \(21.2\), which is not among the given options.

Key concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry to find the length between two points in the coordinate plane. If you have two points with coordinates \(x_1, y_1\) and \(x_2, y_2\), you can calculate the distance between them using:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula derives from the Pythagorean Theorem, where the distance represents the hypotenuse of a right triangle formed by the x and y differences between the points.
In our original exercise, we applied this formula to calculate the lengths of the sides of the quadrilateral formed by the points given. Each calculation involved finding differences in x and y coordinates, squaring those differences, summing them up, and then taking the square root. This approach helped in determining that the shape is a parallelogram due to equal opposite sides.

Parallelogram Properties

A parallelogram is a quadilateral with opposite sides that are equal in length and parallel. This type of geometric shape possesses specific properties:

• Opposite sides are equal and parallel.
• Opposite angles are equal.
• Diagonals bisect each other.
• The adjacent angles are supplementary.

In the original exercise, we confirmed that the given quadrilateral is a parallelogram by verifying that the opposite sides were equal. Once we established this, it allowed us to use parallelogram properties to simplify our further calculations, particularly when it came to finding the area. Understanding these properties is essential, as it guides how we can calculate other aspects of the shape, such as its area.

Equation of a Line

The equation of a line in a coordinate plane is typically expressed in the slope-intercept form: \(y = mx + b\). To determine this equation, you need two pieces of information: the slope \(m\) and the y-intercept \(b\). The slope is the rate at which the y-coordinate changes with respect to the x-coordinate.
We calculate the slope using the difference in y-coordinates divided by the difference in x-coordinates from two given points. For example, in the exercise, the slope was found for line \(AB\) with points \((-4, 2)\) and \(1, -1\):\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 2}{1 + 4} = -\frac{3}{5}\]The slope helps in deriving the full equation of the line. Understanding this concept is key for tasks like drawing altitudes or projecting points on lines in further calculations.

Coordinate Geometry

Coordinate geometry blends geometry and algebra to study the properties and relations of geometric figures within a coordinate system. By using coordinates, one can define geometrical shapes, calculate distances, midpoints, and areas with precision.

• It assigns coordinates to vertices of shapes to find lengths, slopes, and equations.
• Helps find intersections of lines and areas of complex shapes, such as quadrilaterals.

Coordinate geometry allows for the effective graphical representation and transformation of shapes. In our exercise, it involved placing the points in the plane, determining sides' lengths via the distance formula, confirming parallelogram properties, and finding line equations. These steps underlined fundamental concepts and their relationships, leading to solving the quadrilateral's area.