Problem 7 COORDINATE GEOMETRY Given each s... [FREE SOLUTION]

Chapter 6: Problem 7

COORDINATE GEOMETRY Given each set of vertices, determine whether DMNPQ is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. $$M(-4,0), N(-3,3), P(2,2), Q(1,-1)$$

Short Answer

Expert verified

DMNPQ is a rectangle.

Step by step solution

Determine if the Figure is a Rhombus

First, calculate the distance between each pair of adjacent vertices using the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For each pair:- **MN**: \[d = \sqrt{(-3 - (-4))^2 + (3 - 0)^2} = \sqrt{1^2 + 3^2} = \sqrt{10}\]- **NP**: \[d = \sqrt{(2 - (-3))^2 + (2 - 3)^2} = \sqrt{5^2 + (-1)^2} = \sqrt{26}\]- **PQ**: \[d = \sqrt{(1 - 2)^2 + (-1 - 2)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{10}\]- **QM**: \[d = \sqrt{(-4 - 1)^2 + (0 - (-1))^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{26}\]Since opposite sides are equal (MN = PQ and NP = QM), it's not a rhombus because all sides need to be equal for a figure to be a rhombus.

Check if the Figure is a Rectangle

Calculate the slopes of opposite sides to check for parallelism and perpendicularity:- **Slope of MN**: \[\frac{3 - 0}{-3 - (-4)} = 3\]- **Slope of PQ**: \[\frac{-1 - 2}{1 - 2} = 3\]- **Slope of NP**: \[\frac{2 - 3}{2 - (-3)} = -\frac{1}{5}\]- **Slope of QM**: \[\frac{0 - (-1)}{-4 - 1} = -\frac{1}{5}\]Since MN $||$ PQ and NP $||$ QM and slopes of adjacent sides are negative reciprocals (they multiply to -1), the figure must be a rectangle.

Determine if the Figure is a Square

For the figure to be a square, it must be a rectangle with all sides of equal length. From Step 1, we know not all sides are equal ($\sqrt{10} eq \sqrt{26}$), hence the figure cannot be a square.

Conclusion: Classify the Figure

The figure DMNPQ is a rectangle based on the equal lengths of opposite sides and perpendicularity of adjacent sides. It is not a rhombus or a square because all side lengths are not equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula

The distance formula is a fundamental concept in coordinate geometry, used to calculate the distance between two points in a plane. It's derived from the Pythagorean theorem and helps us measure how far apart two points are.

Given two points, $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between these points is calculated as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This formula is particularly useful when analyzing geometric figures, such as when determining if a shape is a rhombus, rectangle, or square based on the length of its sides.

When all sides have the same length, we have a rhombus or a square.
If only the opposite sides are equal, it may suggest a rectangle.

By applying the distance formula to each pair of points, you can compare the lengths to ascertain these geometric properties.

Slope of a Line

The slope of a line is a measure of its steepness or inclination. In coordinate geometry, it tells us how much a line rises or falls as we move from left to right across the graph.

The slope between any two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Key insights regarding slope include:

A positive slope indicates the line is rising.
A negative slope indicates the line is falling.
Zero slope indicates a horizontal line; undefined slope indicates a vertical line.

Analyzing the slopes of different parts of a shape helps in classifying it. For example:

If opposite sides of a quadrilateral have equal slopes, they are parallel.
If adjacent sides have slopes that are negative reciprocals of each other, the lines are perpendicular.

These properties are essential in identifying rectangles, as they require pairs of opposite sides to be parallel and adjacent sides to be perpendicular.

Properties of Quadrilaterals

Quadrilaterals are four-sided polygons, and they have varied properties depending on their specific types, such as rectangles, rhombuses, and squares.

To determine the type of a quadrilateral, you can use its side lengths, angles, and other geometric properties:

**Rectangle**: It has opposite sides parallel and equal, and all angles are right angles. This occurs when the slopes of adjacent sides are negative reciprocals.
**Rhombus**: It requires all four sides to be equal, with opposite sides parallel. Here, distances need to be consistently equal across sides.
**Square**: It combines the properties of a rectangle and a rhombus, sharing equal sides and right angles.

By combining calculations from the distance formula and analyzing slopes, you can deduce these properties. In our example, using these methods helped identify that DMNPQ is a rectangle, as the opposite sides are equal and parallel, and adjacent slopes are negative reciprocals, confirming perpendicularity.

91影视

COORDINATE GEOMETRY Given each set of vertices, determine whether DMNPQ is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. $$M(-4,0), N(-3,3), P(2,2), Q(1,-1)$$

Short Answer

Step by step solution

Determine if the Figure is a Rhombus

Check if the Figure is a Rectangle

Determine if the Figure is a Square

Conclusion: Classify the Figure

Key Concepts

Distance Formula

Slope of a Line

Properties of Quadrilaterals

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