Video: Using slope to Determine Whether a Parallelogram is a Rectangle

A parallelogram has vertices at the coordinates 𝐴(βˆ’1, 2), 𝐡(0, 4), 𝐢(3, 1), and 𝐷(2, βˆ’1). (a) Work out the slope of 𝐴𝐡. (b) Work out the slope of 𝐡𝐢. (c) Work out the product of the slopes from parts (a) and (b). (d) Is 𝐴𝐡𝐢𝐷 a rectangle?

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Video Transcript

A parallelogram has vertices at the coordinates 𝐴 negative one, two; 𝐡 zero, four; 𝐢 three, one; and 𝐷 two, negative one. Part a), work out the slope of 𝐴𝐡. Part b), work out the slope of 𝐡𝐢. Part c), work out the product of the slopes from parts a and b. Part d), is 𝐴𝐡𝐢𝐷 a rectangle?

The gradient or slope between any two points can be calculated using the formula 𝑦 two minus 𝑦 one divided by π‘₯ two minus π‘₯ one. This is the change in 𝑦-coordinates over the change in the π‘₯-coordinates, often known as the rise over the run. In part a, we need to calculate the slope of 𝐴𝐡. We will call the coordinates at point 𝐴 π‘₯ one, 𝑦 one and the coordinates of point 𝐡 π‘₯ two, 𝑦 two. Substituting in these values tells us that the slope of 𝐴𝐡 is equal to four minus two divided by zero minus negative one. Four minus two is equal to two. And zero minus negative one is equal to one, as the two negatives become a positive. Two divided by one is equal to two. This means that the slope of 𝐴𝐡 is equal to two.

The second part of the question wants us to work out the slope of 𝐡𝐢. This time we’ll call point 𝐡 π‘₯ one, 𝑦 one and point 𝐢 π‘₯ two, 𝑦 two. Substituting these values into our formula gives us one minus four divided by three minus zero. One minus four is equal to negative three. And three minus zero is equal to three. Dividing negative three by three gives us an answer of negative one. This means that the slope of 𝐡𝐢 is equal to negative one. The third part of the question asked us to work out the product of parts 𝐴 and 𝐡. The word product means multiply, so we need to multiply two by negative one. Two multiplied by negative one is equal to negative two, as multiplying a positive number by a negative number gives us a negative answer. The product of the slopes from parts 𝐴 and 𝐡 is negative two.

The final part of the question asked us to decide whether 𝐴𝐡𝐢𝐷 is also a rectangle. The length and width of a rectangle meet at right angles. This means that sides 𝐴𝐡 and 𝐡𝐢 are perpendicular. If two slopes are perpendicular, we know that they have a product of negative one. As the product of the slopes 𝐴𝐡 and 𝐡𝐢 was equal to negative two, they will not be perpendicular. We can, therefore, conclude that the parallelogram with coordinates negative one, two; zero, four; three, one; and two, negative one is not a rectangle.

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