The Coordinates of the Vertices of Quadrilateral RSTU

Oct 13, 2025 by windailys.com 34 views

Introduction

In geometry, a quadrilateral is a four-sided polygon. To determine whether a quadrilateral is a rectangle, we need to check if it has four right angles and opposite sides of equal length. In this article, we will explore the coordinates of the vertices of quadrilateral RSTU and determine whether it is a rectangle or not.

The Coordinates of the Vertices of Quadrilateral RSTU

The coordinates of the vertices of quadrilateral RSTU are given as:

R(-4, 1)
S(4, -1)
T(3, -6)
U(-5, -4)

Calculating the Length of the Sides of Quadrilateral RSTU

To determine whether quadrilateral RSTU is a rectangle, we need to calculate the length of its sides. We can use the distance formula to find the length of each side.

Distance Formula

The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Calculating the Length of Side RS

To calculate the length of side RS, we need to find the distance between points R(-4, 1) and S(4, -1).

import math
x1, y1 = -4, 1
x2, y2 = 4, -1
rs = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)
print("The length of side RS is:", rs)

Calculating the Length of Side ST

To calculate the length of side ST, we need to find the distance between points S(4, -1) and T(3, -6).

# Define the coordinates of points S and T
x1, y1 = 4, -1
x2, y2 = 3, -6
st = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)
print("The length of side ST is:", st)

Calculating the Length of Side TU

To calculate the length of side TU, we need to find the distance between points T(3, -6) and U(-5, -4).

# Define the coordinates of points T and U
x1, y1 = 3, -6
x2, y2 = -5, -4
tu = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)
print("The length of side TU is:", tu)

Calculating the Length of Side RU

To calculate the length of side RU, we need to find the distance between points R(-4, 1) and U(-5, -4).

# Define the coordinates of points R and U
x1, y1 = -4, 1
x2, y2 = -5, -4
ru = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)
print("The length of side RU is:", ru)

Determining Whether Quadrilateral RSTU is a Rectangle

To determine whether quadrilateral RSTU is a rectangle, we need to check if it has four right angles and opposite sides of equal length.

Checking for Right Angles

To check for right angles, we need to calculate the slopes of the sides of the quadrilateral.

Slope Formula

The slope formula is given by:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Calculating the Slope of Side RS

To calculate the slope of side RS, we need to find the slope between points R(-4, 1) and S(4, -1).

# Define the coordinates of points R and S
x1, y1 = -4, 1
x2, y2 = 4, -1
m_rs = (y2 - y1) / (x2 - x1)
print("The slope of side RS is:", m_rs)

Calculating the Slope of Side ST

To calculate the slope of side ST, we need to find the slope between points S(4, -1) and T(3, -6).

# Define the coordinates of points S and T
x1, y1 = 4, -1
x2, y2 = 3, -6
m_st = (y2 - y1) / (x2 - x1)
print("The slope of side ST is:", m_st)

Calculating the Slope of Side TU

To calculate the slope of side TU, we need to find the slope between points T(3, -6) and U(-5, -4).

# Define the coordinates of points T and U
x1, y1 = 3, -6
x2, y2 = -5, -4
m_tu = (y2 - y1) / (x2 - x1)
print("The slope of side TU is:", m_tu)

Calculating the Slope of Side RU

To calculate the slope of side RU, we need to find the slope between points R(-4, 1) and U(-5, -4).

# Define the coordinates of points R and U
x1, y1 = -4, 1
x2, y2 = -5, -4
m_ru = (y2 - y1) / (x2 - x1)
print("The slope of side RU is:", m_ru)

Checking for Opposite Sides of Equal Length

To check for opposite sides of equal length, we need to compare the lengths of the sides of the quadrilateral.

# Compare the lengths of the sides of the quadrilateral
if rs == st and tu == ru:
    print("The quadrilateral has opposite sides of equal length.")
else:
    print("The quadrilateral does not have opposite sides of equal length.")

Conclusion

In conclusion, to determine whether quadrilateral RSTU is a rectangle, we need to check if it has four right angles and opposite sides of equal length. We can use the distance formula to calculate the length of the sides of the quadrilateral and the slope formula to calculate the slopes of the sides. By comparing the lengths and slopes of the sides, we can determine whether the quadrilateral is a rectangle or not.

The Final Answer

Based on the calculations, we can see that the quadrilateral RSTU has opposite sides of equal length, but it does not have four right angles. Therefore, the correct statement is:

Quadrilateral RSTU is not a rectangle because it does not have four right angles.

Introduction

In our previous article, we explored the coordinates of the vertices of quadrilateral RSTU and determined whether it is a rectangle or not. In this article, we will answer some frequently asked questions about quadrilateral RSTU.

Q: What is the definition of a rectangle?

A: A rectangle is a quadrilateral with four right angles and opposite sides of equal length.

Q: How do I determine whether a quadrilateral is a rectangle?

A: To determine whether a quadrilateral is a rectangle, you need to check if it has four right angles and opposite sides of equal length. You can use the distance formula to calculate the length of the sides of the quadrilateral and the slope formula to calculate the slopes of the sides.

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Q: What is the slope formula?

A: The slope formula is a mathematical formula used to calculate the slope of a line passing through two points in a coordinate plane. It is given by:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of two points.

Q: How do I calculate the length of the sides of a quadrilateral?

A: To calculate the length of the sides of a quadrilateral, you need to use the distance formula. You can use the distance formula to calculate the length of each side of the quadrilateral.

Q: How do I calculate the slopes of the sides of a quadrilateral?

A: To calculate the slopes of the sides of a quadrilateral, you need to use the slope formula. You can use the slope formula to calculate the slope of each side of the quadrilateral.

Q: What is the significance of opposite sides of equal length in a rectangle?

A: Opposite sides of equal length are a characteristic of a rectangle. If a quadrilateral has opposite sides of equal length, it is likely to be a rectangle.

Q: What is the significance of four right angles in a rectangle?

A: Four right angles are a characteristic of a rectangle. If a quadrilateral has four right angles, it is likely to be a rectangle.

Q: Can a quadrilateral have both opposite sides of equal length and four right angles?

A: Yes, a quadrilateral can have both opposite sides of equal length and four right angles. In this case, the quadrilateral is a rectangle.

Q: Can a quadrilateral have opposite sides of equal length but not four right angles?

A: Yes, a quadrilateral can have opposite sides of equal length but not four right angles. In this case, the quadrilateral is not a rectangle.

Q: Can a quadrilateral have four right angles but not opposite sides of equal length?

A: Yes, a quadrilateral can have four right angles but not opposite sides of equal length. In this case, the quadrilateral is not a rectangle.

Conclusion

In conclusion, determining whether a quadrilateral is a rectangle or not requires checking if it has four right angles and opposite sides of equal length. We can use the distance formula to calculate the length of the sides of the quadrilateral and the slope formula to calculate the slopes of the sides. By answering these frequently asked questions, we can gain a better understanding of quadrilateral RSTU and its properties.

Frequently Asked Questions

What is the definition of a rectangle?
How do I determine whether a quadrilateral is a rectangle?
What is the distance formula?
What is the slope formula?
How do I calculate the length of the sides of a quadrilateral?
How do I calculate the slopes of the sides of a quadrilateral?
What is the significance of opposite sides of equal length in a rectangle?
What is the significance of four right angles in a rectangle?
Can a quadrilateral have both opposite sides of equal length and four right angles?
Can a quadrilateral have opposite sides of equal length but not four right angles?
Can a quadrilateral have four right angles but not opposite sides of equal length?

Answers

A rectangle is a quadrilateral with four right angles and opposite sides of equal length.
To determine whether a quadrilateral is a rectangle, you need to check if it has four right angles and opposite sides of equal length.
The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane.
The slope formula is a mathematical formula used to calculate the slope of a line passing through two points in a coordinate plane.
To calculate the length of the sides of a quadrilateral, you need to use the distance formula.
To calculate the slopes of the sides of a quadrilateral, you need to use the slope formula.
Opposite sides of equal length are a characteristic of a rectangle.
Four right angles are a characteristic of a rectangle.
Yes, a quadrilateral can have both opposite sides of equal length and four right angles.
Yes, a quadrilateral can have opposite sides of equal length but not four right angles.
Yes, a quadrilateral can have four right angles but not opposite sides of equal length.