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Chapter 11: Problem 6

COORDINATE GEOMETRY Find the area of each figure given the coordinates of the vertices. rhombus \(L M P Q\) with \(L(-4,3), M(-2,4), P(0,3),\) and \(Q(-2,2)\)

Short Answer

Expert verified
The area of rhombus LMNP is 4 square units.

Step by step solution

01

Understand the Problem

We need to find the area of a rhombus given its vertices' coordinates in a coordinate plane. The four vertices are L(-4, 3), M(-2, 4), P(0, 3), and Q(-2, 2).
02

Verify the Shape

To confirm it's a rhombus, remember that a rhombus is a quadrilateral where *all sides are equal in length*. We calculate the distance between consecutive vertices to verify equality. Each side should have the same length.
03

Distance Formula

Use the distance formula to find the length of each side: Distance \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).Calculate LM, MP, PQ, and QL.- For LM: \[ \sqrt{(-2 - (-4))^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{5} \]- For MP: \[ \sqrt{(0 - (-2))^2 + (3 - 4)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{5} \]- For PQ: \[ \sqrt{(0 - (-2))^2 + (3 - 2)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{5} \]- For QL: \[ \sqrt{(-2 - (-4))^2 + (2 - 3)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{5} \]
04

Diagonal Lengths

The area of a rhombus can be calculated using the lengths of its diagonals. Calculate the lengths of diagonal LP and diagonal MQ. Use the distance formula for these calculations.- For LP: \[ \sqrt{(0 - (-4))^2 + (3 - 3)^2} = \sqrt{(4)^2 + (0)^2} = 4 \]- For MQ: \[ \sqrt{(0 - (-2))^2 + (3 - 2)^2} = \sqrt{(2)^2 + (0)^2} = 2 \]
05

Area Formula for Rhombus

The area of a rhombus can be found using the formula: \( A = \frac{1}{2} \times \text{(Diagonal 1)} \times \text{(Diagonal 2)} \).Substitute the lengths of LP and MQ into the formula:\[ A = \frac{1}{2} \times 4 \times 2 = 4 \]
06

Conclusion

The area of the rhombus LMNP, given the vertices, is 4 square units.

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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 tool in coordinate geometry. It helps determine the distance between two points in a plane based on their coordinates. This formula is derived from the Pythagorean theorem and is given by:
  • Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
To apply this in our exercise, we checked the distances between all consecutive vertices of the rhombus. The calculations showed each side having a length of \\sqrt{5} \, confirming the shape as a rhombus because all sides are equal. This concept ensures that we can verify shapes and measure distances accurately between points on coordinate planes.
Area of a Rhombus
For a rhombus, area calculation can be simplified using its diagonals. The formula for finding the area of a rhombus is:
  • Area \( A = \frac{1}{2} \times \text{(Diagonal 1)} \times \text{(Diagonal 2)} \)
This formula is particularly practical when the vertices are given, as in this exercise. To use it, we first calculated the length of each diagonal using the distance formula: the length of diagonal LP is 4 units, and MQ is 2 units. Substituting these values into the area formula gives us an area of 4 square units. This approach simplifies the process of finding the area by avoiding more complex methods that require side length measurements or angles.
Diagonals in a Quadrilateral
Diagonals play a crucial role in understanding and calculating properties of quadrilaterals. In a rhombus, the diagonals not only bisect each other but also play a critical role in calculating area.
  • Diagonals of a rhombus are perpendicular to each other.
  • They are essential in verifying the nature of the given quadrilateral.
  • For any quadrilateral, diagonals help divide the shape into manageable triangles.
In our example, the diagonals LP and MQ allowed us to apply the area formula effectively. This insight highlights the importance of identifying and calculating diagonal lengths to utilize geometric properties, especially in coordinate geometry, to make mathematical deductions more accessible and insightful.

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Most popular questions from this chapter

Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram. Then find the area of the quadrilateral. $$A(-3,2), B(4,2), C(2,-1), D(-5,-1)$$

Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram. Then find the perimeter and area of the quadrilateral. $$E(-5,-3), F(3,-3), G(5,4), H(-3,4)$$

Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram. Then find the perimeter and area of the quadrilateral. $$R(-2,4), S(8,4), T(8,-3), U(-2,-3)$$

GARDENS For Exercises 32 and \(33,\) use the following information. Keisha designed a garden that is shaped like two congruent rhombi. She wants the long diagonals lined with a stone walkway. The total area of the garden is 150 square feet, and the shorter diagonals are each 12 feet long. Find the length of each side of the garden. (IMAGE CAN'T COPY)

GARDENS For Exercises 32 and \(33,\) use the following information. Keisha designed a garden that is shaped like two congruent rhombi. She wants the long diagonals lined with a stone walkway. The total area of the garden is 150 square feet, and the shorter diagonals are each 12 feet long. Find the length of each stone walkway. (IMAGE CAN'T COPY)
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