Select The Correct Answer.The Vertices Of A Parallelogram Are { A\left(x_1, Y_1\right), B\left(x_2, Y_2\right), C\left(x_3, Y_3\right), $}$ And { D\left(x_4, Y_4\right) $}$. Which Of The Following Must Be True If Parallelogram

Mar 12, 2025 by ADMIN 227 views

**Understanding the Properties of a Parallelogram in Coordinate Geometry**

Introduction

In coordinate geometry, a parallelogram is a quadrilateral with opposite sides that are parallel to each other. The vertices of a parallelogram are denoted by the points A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4). In this article, we will explore the properties of a parallelogram and determine which of the given statements must be true.

Properties of a Parallelogram

A parallelogram has several properties that can be used to determine its characteristics. Some of the key properties of a parallelogram include:

Opposite sides are parallel: The opposite sides of a parallelogram are parallel to each other. This means that the slope of the line segment AB is equal to the slope of the line segment CD.
Opposite sides are equal: The opposite sides of a parallelogram are equal in length. This means that the distance between points A and B is equal to the distance between points C and D.
Diagonals bisect each other: The diagonals of a parallelogram bisect each other. This means that the point of intersection of the diagonals is the midpoint of both diagonals.

Must be True Statements

Now that we have discussed the properties of a parallelogram, let's examine the given statements and determine which one must be true.

Statement 1: The slope of the line segment AB is equal to the slope of the line segment CD

This statement is true because opposite sides of a parallelogram are parallel. The slope of the line segment AB is equal to the slope of the line segment CD.

Statement 2: The distance between points A and B is equal to the distance between points C and D

This statement is true because opposite sides of a parallelogram are equal in length. The distance between points A and B is equal to the distance between points C and D.

Statement 3: The point of intersection of the diagonals is the midpoint of both diagonals

This statement is true because the diagonals of a parallelogram bisect each other. The point of intersection of the diagonals is the midpoint of both diagonals.

Statement 4: The sum of the x-coordinates of points A and C is equal to the sum of the x-coordinates of points B and D

This statement is not necessarily true. The sum of the x-coordinates of points A and C is not necessarily equal to the sum of the x-coordinates of points B and D.

Statement 5: The sum of the y-coordinates of points A and C is equal to the sum of the y-coordinates of points B and D

This statement is not necessarily true. The sum of the y-coordinates of points A and C is not necessarily equal to the sum of the y-coordinates of points B and D.

Conclusion

In conclusion, the statements that must be true if a parallelogram is given are:

The slope of the line segment AB is equal to the slope of the line segment CD.
The distance between points A and B is equal to the distance between points C and D.
The point of intersection of the diagonals is the midpoint of both diagonals.

These statements are true because they are based on the properties of a parallelogram. The other statements are not necessarily true and may not be applicable to all parallelograms.

Example Problems

Here are some example problems to help illustrate the concepts discussed in this article.

Problem 1

Given the vertices of a parallelogram A(1, 2), B(3, 4), C(5, 6), and D(7, 8), determine the slope of the line segment AB and the slope of the line segment CD.

Solution

The slope of the line segment AB is (4 - 2) / (3 - 1) = 2 / 2 = 1.

The slope of the line segment CD is (6 - 8) / (5 - 7) = -2 / -2 = 1.

Problem 2

Given the vertices of a parallelogram A(1, 2), B(3, 4), C(5, 6), and D(7, 8), determine the distance between points A and B and the distance between points C and D.

Solution

The distance between points A and B is sqrt((3 - 1)^2 + (4 - 2)^2) = sqrt(4 + 4) = sqrt(8).

The distance between points C and D is sqrt((7 - 5)^2 + (8 - 6)^2) = sqrt(4 + 4) = sqrt(8).

Problem 3

Given the vertices of a parallelogram A(1, 2), B(3, 4), C(5, 6), and D(7, 8), determine the point of intersection of the diagonals.

Solution

The point of intersection of the diagonals is the midpoint of both diagonals. The midpoint of the diagonal AC is ((1 + 5) / 2, (2 + 6) / 2) = (3, 4).

The midpoint of the diagonal BD is ((3 + 7) / 2, (4 + 8) / 2) = (5, 6).

The point of intersection of the diagonals is (3, 4).

Final Thoughts

In conclusion, the properties of a parallelogram are an essential part of coordinate geometry. Understanding the properties of a parallelogram can help us determine the characteristics of a given figure and solve problems involving parallelograms. The statements that must be true if a parallelogram is given are the slope of the line segment AB is equal to the slope of the line segment CD, the distance between points A and B is equal to the distance between points C and D, and the point of intersection of the diagonals is the midpoint of both diagonals.

Introduction

In our previous article, we discussed the properties of a parallelogram and determined which statements must be true if a parallelogram is given. In this article, we will answer some frequently asked questions about parallelograms.

Q1: What is the difference between a parallelogram and a rectangle?

A1: A parallelogram is a quadrilateral with opposite sides that are parallel to each other. A rectangle is a quadrilateral with opposite sides that are equal in length and parallel to each other. While all rectangles are parallelograms, not all parallelograms are rectangles.

Q2: What is the formula for the area of a parallelogram?

A2: The formula for the area of a parallelogram is base × height. The base is the length of one of the sides of the parallelogram, and the height is the distance between the base and the opposite side.

Q3: How do you find the midpoint of a diagonal of a parallelogram?

A3: To find the midpoint of a diagonal of a parallelogram, you can use the midpoint formula: (x1 + x2) / 2, (y1 + y2) / 2.

Q4: What is the relationship between the diagonals of a parallelogram?

A4: The diagonals of a parallelogram bisect each other. This means that the point of intersection of the diagonals is the midpoint of both diagonals.

Q5: Can a parallelogram have a right angle?

A5: Yes, a parallelogram can have a right angle. In fact, a parallelogram with a right angle is called a rectangle.

Q6: How do you determine if a quadrilateral is a parallelogram?

A6: To determine if a quadrilateral is a parallelogram, you can use the following criteria:

Check if the opposite sides are parallel.
Check if the opposite sides are equal in length.
Check if the diagonals bisect each other.

If all of these criteria are met, then the quadrilateral is a parallelogram.

Q7: Can a parallelogram have a side that is a diameter of a circle?

A7: No, a parallelogram cannot have a side that is a diameter of a circle. This is because a diameter of a circle is a line segment that passes through the center of the circle and has endpoints on the circle. A parallelogram, on the other hand, is a quadrilateral with opposite sides that are parallel to each other.

Q8: How do you find the slope of a diagonal of a parallelogram?

A8: To find the slope of a diagonal of a parallelogram, you can use the slope formula: (y2 - y1) / (x2 - x1).

Q9: Can a parallelogram have a side that is a tangent to a circle?

A9: Yes, a parallelogram can have a side that is a tangent to a circle. In fact, a parallelogram with a side that is a tangent to a circle is called a cyclic quadrilateral.

Q10: How do you determine if a quadrilateral is a cyclic quadrilateral?

A10: To determine if a quadrilateral is a cyclic quadrilateral, you can use the following criteria:

Check if the quadrilateral has a side that is a tangent to a circle.
Check if the opposite sides are parallel.
Check if the opposite sides are equal in length.
Check if the diagonals bisect each other.

If all of these criteria are met, then the quadrilateral is a cyclic quadrilateral.

Conclusion

In conclusion, we have answered some frequently asked questions about parallelograms. We hope that this article has been helpful in clarifying some of the concepts related to parallelograms. If you have any further questions, please don't hesitate to ask.