Question

Geometry Use slopes to show that the quadrilateral whose vertices are \((-1,0),(2,3),(1,-2),\) and (4,1) is a rectangle.

Short answer

The quadrilateral is a rectangle because opposite sides are parallel and adjacent sides are perpendicular.

Step-by-step solution

Find the Slopes of Opposite Sides

First, calculate the slopes of the line segments between the given vertices to determine if opposite sides are parallel. Use the slope formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).

Compute the Slope of \((-1,0)\text{ and }(2,3)\)

Calculate the slope between \((-1,0)\text{ and }(2,3)\): \(\text{slope}_{AB} = \frac{3-0}{2-(-1)} = \frac{3}{3} = 1\).

Compute the Slope of \((1,-2)\text{ and }(4,1)\)

Calculate the slope between \((1,-2)\text{ and }(4,1)\): \(\text{slope}_{CD} = \frac{1-(-2)}{4-1} = \frac{3}{3} = 1\).

Compute the Slope of \((2,3)\text{ and }(4,1)\)

Calculate the slope between \((2,3)\text{ and }(4,1)\): \(\text{slope}_{BC} = \frac{1-3}{4-2} = \frac{-2}{2} = -1\).

Compute the Slope of \((-1,0)\text{ and }(1,-2)\)

Calculate the slope between \((-1,0)\text{ and }(1,-2)\): \(\text{slope}_{AD} = \frac{-2-0}{1-(-1)} = \frac{-2}{2} = -1\).

Verify Opposite Sides

Note that \(\text{slope}_{AB} = \text{slope}_{CD} = 1\) and \(\text{slope}_{BC} = \text{slope}_{AD} = -1\). Since opposite sides have equal slopes, they are parallel.

Confirm Perpendicularity

The slopes of adjacent sides \((1 \text{ and } -1)\) are negative reciprocals. Therefore, the sides are perpendicular.

Conclusion

Since opposite sides are parallel and adjacent sides are perpendicular, the quadrilateral is a rectangle.

Key concepts

slopes and line segments

Understanding slopes and line segments is crucial in geometry. The slope of a line segment shows how steep the line is and in which direction it goes. We calculate the slope using the formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).
To apply this, take two points on a line. For example, if you have points A \((-1,0)\) and B \((2,3)\), plug these values into the formula to find the slope: \(\text{slope}_{AB} = \frac{3-0}{2-(-1)} = 1\).
This shows that the line segment AB rises by 1 unit for each unit it moves to the right.
Calculating slopes for various line segments helps us understand their relationship with each other.

quadrilaterals

A quadrilateral is any four-sided polygon. These shapes can have various properties and forms.
Common types of quadrilaterals include:

• Squares
• Rectangles
• Rhombuses
• Parallelograms

To classify a quadrilateral, we look at its sides, angles, and sometimes its diagonals.
In the given problem, we identified a quadrilateral by its vertices \((-1,0),(2,3),(1,-2),(4,1)\).
We then used slopes to understand how these sides interact.
Specifically, for rectangles, opposite sides must be parallel and adjacent sides perpendicular.

parallelograms

A parallelogram is a special type of quadrilateral. It has two pairs of parallel sides. To identify a parallelogram, check the slopes of opposite sides using the slope formula.
In our exercise, we found the slopes of opposite sides \(( \text{slope}_{AB} = 1 \text{ and } \text{slope}_{CD} = 1 )\) confirming that AB is parallel to CD. Similarly, \(\text{slope}_{BC} = -1 \text{ and } \text{slope}_{AD} = -1 )\) showed BC is parallel to AD.
Additionally, for rectangles (a type of parallelogram), adjacent sides must be perpendicular. A slope of 1 and \(-1\) are negative reciprocals indicating perpendicularity. Thus, our quadrilateral is not just a parallelogram but a rectangle.