Consider a quadrilateral with vertices A(2, 2), B(2,5), C(7, 2), and D(7,5). What is the shape and area of quadrilateral ABCD? Quadrilateral ABCD is a ______ with an area of ______ square units.
Answer (2)
Calculate the lengths of the sides: A B = 3 , BC = 34 , C D = 3 , D A = 34 .
Calculate the slopes of the sides to determine that AB and CD are vertical lines and AC is a horizontal line, indicating perpendicularity.
Determine that the quadrilateral is a rectangle.
Calculate the area of the rectangle: A re a = 3 × 5 = 15 square units. The final answer is 15 .
Explanation
Analyze the vertices Let's analyze the given vertices of the quadrilateral ABCD: A(2, 2), B(2, 5), C(7, 2), and D(7, 5). We need to determine the shape of the quadrilateral and calculate its area.
Calculate side lengths First, let's calculate the lengths of the sides using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Length of AB: A B = ( 2 − 2 ) 2 + ( 5 − 2 ) 2 = 0 2 + 3 2 = 9 = 3 .
Length of BC: BC = ( 7 − 2 ) 2 + ( 2 − 5 ) 2 = 5 2 + ( − 3 ) 2 = 25 + 9 = 34 .
Length of CD: C D = ( 7 − 7 ) 2 + ( 2 − 5 ) 2 = 0 2 + ( − 3 ) 2 = 9 = 3 .
Length of DA: D A = ( 2 − 7 ) 2 + ( 2 − 5 ) 2 = ( − 5 ) 2 + ( − 3 ) 2 = 25 + 9 = 34 .
We observe that AB = CD = 3 and BC = DA = 34 . This suggests that the quadrilateral is a rectangle or a parallelogram.
Calculate slopes Now, let's calculate the slopes of the sides using the slope formula: m = x 2 − x 1 y 2 − y 1 .
Slope of AB: m A B = 2 − 2 5 − 2 = 0 3 . This is undefined, which means AB is a vertical line. Slope of BC: m BC = 7 − 2 2 − 5 = 5 − 3 = − 5 3 .
Slope of CD: m C D = 7 − 7 5 − 2 = 0 3 . This is undefined, which means CD is a vertical line. Slope of DA: m D A = 7 − 2 5 − 2 = 5 3 .
Since AB and CD have undefined slopes, they are parallel. However, BC and DA do not have the same slope, so they are not parallel. This means that the quadrilateral is not a parallelogram. Let's recalculate the slope of DA: m D A = 2 − 7 2 − 5 = − 5 − 3 = 5 3 .
Since AB and CD are vertical lines, they are parallel. Also, the slope of BC is − 5 3 and the slope of DA is 5 3 . These are not the same, so BC and DA are not parallel. However, notice that the x-coordinates of A and B are the same, and the x-coordinates of C and D are the same. Also, the y-coordinates of A and C are the same, and the y-coordinates of B and D are the same. This means that AB and CD are vertical lines, and AC and BD are horizontal lines. Therefore, the quadrilateral is a rectangle.
Determine the shape and calculate the area Since AB is a vertical line and AC is a horizontal line, they are perpendicular to each other. Therefore, the quadrilateral is a rectangle.
To find the area of the rectangle, we need to find the lengths of two adjacent sides. We already know that AB = 3. We can find the length of AC using the distance formula: A C = ( 7 − 2 ) 2 + ( 2 − 2 ) 2 = 5 2 + 0 2 = 25 = 5 .
Therefore, the area of the rectangle is A re a = A B × A C = 3 × 5 = 15 square units.
State the final answer The quadrilateral ABCD is a rectangle with an area of 15 square units.
Examples
Understanding coordinate geometry and shapes like rectangles is crucial in various real-world applications. For instance, architects use these principles to design buildings and ensure walls are parallel and corners are right angles. Similarly, city planners use coordinate systems to map out streets and blocks, ensuring efficient layouts. Even in creating digital graphics or video games, knowing how to define and manipulate shapes using coordinates is essential for creating visual elements and environments.
The quadrilateral ABCD is a rectangle with an area of 15 square units.
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