Question

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

Short answer

The quadrilateral is a square because all sides are equal and perpendicular to each other.

Step-by-step solution

- Calculate the Slopes of All Sides

Find the slope between each pair of adjacent vertices to check if the sides are perpendicular: The slope formula is given by: \(m = \frac{y_2-y_1}{x_2-x_1}\).1. Slope of side between (0,0) and (1,3): \(m_1 = \frac{3-0}{1-0} = 3\).2. Slope of side between (1,3) and (4,2): \(m_2 = \frac{2-3}{4-1} = -\frac{1}{3}\).3. Slope of side between (4,2) and (3,-1): \(m_3 = \frac{-1-2}{3-4} = 3\).4. Slope of side between (3,-1) and (0,0): \(m_4 = \frac{0+1}{0-3} = -\frac{1}{3}\).

- Confirm Perpendicularity

Check if adjacent sides are perpendicular by confirming that the product of their slopes equals -1:1. \(m_1 \times m_2 = 3 \times (-\frac{1}{3}) = -1\).2. \(m_2 \times m_3 = (-\frac{1}{3}) \times 3 = -1\).3. \(m_3 \times m_4 = 3 \times (-\frac{1}{3}) = -1\).4. \(m_4 \times m_1 = (-\frac{1}{3}) \times 3 = -1\).Since the product of the slopes of each pair of adjacent sides is -1, all sides are perpendicular.

- Calculate the Length of All Sides Using the Distance Formula

Use the distance formula to find the length of each side:The distance formula is given by: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).1. Distance between (0,0) and (1,3): \(d_1 = \sqrt{(1-0)^2 + (3-0)^2} = \sqrt{1 + 9} = \sqrt{10}\).2. Distance between (1,3) and (4,2): \(d_2 = \sqrt{(4-1)^2 + (2-3)^2} = \sqrt{9 + 1} = \sqrt{10}\).3. Distance between (4,2) and (3,-1): \(d_3 = \sqrt{(3-4)^2 + (-1-2)^2} = \sqrt{1 + 9} = \sqrt{10}\).4. Distance between (3,-1) and (0,0): \(d_4 = \sqrt{(0-3)^2 + (0+1)^2} = \sqrt{9 + 1} = \sqrt{10}\).All sides have equal length, \(\sqrt{10}\).

- Verify the Quadrilateral is a Square

To confirm the quadrilateral is a square, check that all sides have equal length and each pair of adjacent sides are perpendicular:1. All sides are equally \(\sqrt{10}\).2. Each pair of adjacent sides are perpendicular (from Step 2).Hence, the given quadrilateral with vertices (0,0),(1,3),(4,2) and (3,-1) is a square.

Key concepts

Slopes

In geometry, the slope of a line measures how steep the line is. We calculate the slope using the formula: \(m = \frac{y_2-y_1}{x_2-x_1}\). This formula finds the ratio of vertical change (rise) to horizontal change (run) between two points. For example, the slope between points (0,0) and (1,3) is found like this: \(m = \frac{3-0}{1-0} = 3\). Identifying slopes is crucial to understanding the orientation of sides in a quadrilateral and checking if they are perpendicular.

Distance Formula

To find the length of a side in a quadrilateral, we use the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This formula calculates the straight-line distance between two points. For instance, the distance between points (0,0) and (1,3) is: \(d = \sqrt{(1-0)^2 + (3-0)^2} = \sqrt{1 + 9} = \sqrt{10}\). Using the distance formula, you can verify that all sides of a quadrilateral are equal. This is a key step in verifying the properties of a square.

Perpendicular Lines

Two lines are perpendicular when they intersect at a 90-degree angle. In terms of slopes, two lines are perpendicular if the product of their slopes is -1. For instance, if you have a slope of 3 and another slope of \(-\frac{1}{3}\), multiplying them gives: \(3 \times -\frac{1}{3} = -1\). Verifying perpendicularity ensures that all angles in a quadrilateral are right angles, which is a necessary property for a shape to be classified as a square.

Verifying Squares

To verify that a quadrilateral is a square, follow these steps:

• Ensure all four sides are equal using the distance formula.
• Check that each pair of adjacent sides is perpendicular using slopes.

If both conditions are met, as we did with the vertices (0,0), (1,3), (4,2), and (3,-1), you can confidently say the quadrilateral is a square. By confirming both equal side lengths and right angles, you adhere to the geometric definition of a square.