How to Determine the Area of a Rectangle
This video shows how to determine a rectangle. A rectangle is a parallelogram, which means both pairs of opposite sides are parallel, and all angles are right angles.
Determining a Rectangle
Determine if quadrilateral ABCD is a rectangle. A rectangle is a parallelogram, which means both pairs of opposite sides are parallel, and all four angles are right angles. To determine if this quadrilateral, ABCD, is a rectangle, we’ll need to determine if both pairs of opposite sides are parallel and if our four angles are right angles.
We’ll start by determining if the opposite sides are parallel. If the opposite sides are parallel that means that these segments AB and CD must be parallel, and parallel lines have the same slope. We’re going to find the slope of each one of our lines. You can find the slope a couple of different ways.
You could count, slope is rise over run. M is used indicate slop, so you could count on your coordinate plane and find the rise over run, or the change in Y over the change in X. Or you can use the slope formula, which is Y2 minus Y1, divided by X2, minus X1. Since slope is change in Y divided by change in X, I’m just going to count our slope.
If I’m finding the slope of segment AB, I want to find my rise over run. Starting at A, if I’m going from A to B then I have to go down to from 4 to 3 and then down to 2, and then over 1, 2, 3, 4, 5, 6, 7, 8 places to get to B. My rise again was down 2, which is a negative 2, and my run, once I went down 2, I had to go across 8 to get to B.
Negative 2 over 8, which simplifies to be negative one fourth. Now, to find my slop for Segment CD. My slopes for AB and CD must be parallel in order for this to still possibly be a rectangle. If I’m going from D to C then I’m going to rise over run fist, so to get to C I’m going to have to go down 1, 2 so that’s my rise, Negative 2.
Then my run across 1, 2, 3, 4, 5, 6, 7, 8 to get to Point C, which simplifies to be negative 1/4. Since my slopes for segments AB and CD are the same, that means that those lines are parallel. I still have to check my other segments, because just because one pair of sides is parallel doesn’t make that this shape a rectangle.
It could be a trapezoid. A trapezoid is a quadrilateral with just one pair of opposite sides parallel. Now I need to check the slopes for my other line segments AD and BC to see if they are parallel to each other. I’ll find the slope of line AD. Slope Is rise over run, and I’m going to go from D to A, to get from D to A.
I would have to rise 1, 2, 3, 4 and run 1. Rise 4 run 1. The slope for Segment BC to get from C to D I’d have to rise 1, 2, 3, 4 and run 1. Rise 4 run 1. My slopes for segment’s AD and BC are also congruent, which means that these lines segment’s AD and BC are also parallel. Now, we do know that both pairs of are opposite sides are parallel.
That still doesn’t confirm that this is a quadrilateral. All that means is it’s a parallelogram. In order for this shape to be a rectangle, my four angles would need to be right angles. If this is a right angle, then that means that these lines AB and AD are perpendicular to each other.
Perpendicular lines have opposite reciprocal slopes, and opposite reciprocal means opposite, different signs, reciprocal means flip. Lines AB and AD. That’s this slope, AB and AD, this slope. These slopes are opposite reciprocal of each other. Negative 1/4, positive 4/1 which means those lines are perpendicular to each other.
This is a right angle. Now, we’ll check lines AD and DC. The slop for AD is 4/1 and the slope first CD is negative 1/4. Again, these are opposite reciprocal of each other, which means those lines are perpendicular. So this is also a right angle. Lines AB and BC, the slope for AB is negative 1/4.
The slope for BC is 4/1, negative 1/4 and 4/1 are opposite reciprocals of each other. Those lines are perpendicular, so that is also a right angle. Our last two lines are BC and CD. The slope for line BC is 4/1, and the slope for line CD is negative 1/4. 4/1 and negative 1/4 are opposite reciprocals.
Negative, positive, flipped over. That means they are perpendicular to each other, which means they form a right angle. Therefore, quadrilateral ABCD is a rectangle. If our slopes for Lines AB and CD had not been congruity, then that would have meant those lines were not parallel and then it wouldn’t be a rectangle.
It’d be the same thing if our slopes for AD BC weren’t the same, then those lines wouldn’t be parallel, and it would not be a rectangle. Also, for our perpendicular lines all of my lines AB, AD segment AD and segment DC, segments CD and Segment CB, Segment BC and segment BA, all of those segments had to be perpendicular to each other so that all four of our right angles were right angles. If any one of those things was not true then this would not be a rectangle, but they were, so it is.