Question

1-85a. plot the following points on another sheet of graph paper and connect them in the order given. then connect points a and d. click the link to so you can use the online tool to help you graph 1-85 hw etool. a(-3,4), b(1,6), c(5,-2), and d(1,-4) a rectangle is a four-sided polygon with four right angles. does the shape you graphed appear to be a rectangle? choose the best answer. note: your next question is related to this one! keep your online tool open so you can reference it!!! it is not possible to tell if the angles on that quadrilateral are right angles unless you measure them with a protractor. therefore, i cant tell if this is a rectangle or not. yes. the short sides are perpendicular to the long sides. i can tell they are perpendicular by their slopes. this means the shape has four sides and four right angles. it is a rectangle. no. this is not a rectangle. a rectangle has to have sides that are straight up and down. this shape is tipped on its side, so it is not a rectangle. yes. i can tell just by looking at it that it is a rectangle.

Explanation:

To determine if the shape is a rectangle, we calculate the slopes of the sides. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Slope of AB: $\frac{6 - 4}{1 - (-3)} = \frac{2}{4} = \frac{1}{2}$
• Slope of BC: $\frac{-2 - 6}{5 - 1} = \frac{-8}{4} = -2$
• Slope of CD: $\frac{-4 - (-2)}{1 - 5} = \frac{-2}{-4} = \frac{1}{2}$
• Slope of DA: $\frac{4 - (-4)}{-3 - 1} = \frac{8}{-4} = -2$
• Slope of AD: $\frac{-4 - 4}{1 - (-3)} = \frac{-8}{4} = -2$ (wait, no, AD connects A(-3,4) and D(1,-4), so slope is $\frac{-4 - 4}{1 - (-3)} = \frac{-8}{4} = -2$? Wait, no, AB slope is 1/2, BC slope is -2. The product of slopes of AB and BC is (1/2)(-2) = -1, so they are perpendicular. Similarly, BC and CD: (-2)(1/2) = -1, perpendicular. CD and DA: (1/2)(-2) = -1, perpendicular. DA and AB: (-2)(1/2) = -1, perpendicular. Also, opposite sides have equal slopes (AB and CD: 1/2, BC and DA: -2), so opposite sides are parallel. So it's a rectangle. The second option explains this by checking slopes (perpendicularity) and parallel sides, which is correct. The first option is wrong because we can use slopes (no protractor needed). The third is wrong (rectangle doesn't need sides vertical/horizontal). The fourth is wrong (can't just "look"—need to check properties).

Answer:

Yes. The short sides are perpendicular to the long sides. I can tell they are perpendicular by their slopes. This means the shape has four sides and four right angles. It IS a rectangle.