The vertices of quadrilateral ABCD are A(0, 5), B(9, 2), C (7, –4), and D(–2, –1). Is ABCD a rectangle? Explain.
To determine whether ABCD is a rectangle, we need to check if all the angles are right angles (90 degrees) and if the lengths of opposite sides are equal.
Step 1: Find the lengths of all four sides of the quadrilateral ABCD.
To find the length of a side, you can use the distance formula:
Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2)
Let's calculate the lengths of the sides:
- Side AB: d(AB) = sqrt((9-0)^2 + (2-5)^2)
- Side BC: d(BC) = sqrt((7-9)^2 + (-4-2)^2)
- Side CD: d(CD) = sqrt((-2-7)^2 + (-1-(-4))^2)
- Side DA: d(DA) = sqrt((0-(-2))^2 + (5-(-1))^2)
Step 2: Check if all angles are right angles.
To check if the angles are right angles, we can calculate the slopes of consecutive sides. If the slopes are negative reciprocals of each other, the adjacent sides are perpendicular, and hence the angles are right angles.
Slope formula: m = (y2-y1)/(x2-x1)
Let's calculate the slopes:
- Slope of AB: m(AB) = (2-5)/(9-0)
- Slope of BC: m(BC) = (-4-2)/(7-9)
- Slope of CD: m(CD) = (-1-(-4))/(-2-7)
- Slope of DA: m(DA) = (5-(-1))/(0-(-2))
Step 3: Check if the lengths of opposite sides are equal.
To determine if the opposite sides are equal, we compare their lengths:
- Compare AB and CD.
- Compare BC and DA.
If all angles are right angles, the lengths of opposite sides are equal, and we can conclude that ABCD is a rectangle.
Now, let's perform the calculations and make the comparisons.
AB = √(9^2+3^2) = √90
Check the other lengths. If you find that AB=CD and BC=AD, then you have a parallelogram.
AB has slope -1/3
BC has slope 3 so AB⊥BC
Make sure that AD⊥CD