Finding the Missing Corner

Oct 13, 2025 by windailys.com 41 views

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Introduction

In mathematics, a rectangle is a four-sided shape with four right angles. When a rectangle is graphed on the coordinate plane, its vertices (or corners) are represented by points with specific coordinates. Given three of the four corners of a rectangle, we can find the coordinates of the missing corner by using the properties of rectangles.

Properties of Rectangles

A rectangle has several key properties that we can use to find the missing corner:

The opposite sides of a rectangle are equal in length.
The opposite angles of a rectangle are equal.
The diagonals of a rectangle bisect each other.
The diagonals of a rectangle are equal in length.

Given Points

We are given three points that represent three of the four corners of a rectangle:

$(3,2)$
$(0,-4)$
$(-1,4)$

Finding the Missing Corner

To find the missing corner, we need to use the properties of rectangles. Let's analyze the given points:

The point $(3,2)$ has an x-coordinate of 3 and a y-coordinate of 2.
The point $(0,-4)$ has an x-coordinate of 0 and a y-coordinate of -4.
The point $(-1,4)$ has an x-coordinate of -1 and a y-coordinate of 4.

Using the Properties of Rectangles

Since the opposite sides of a rectangle are equal in length, we can use this property to find the missing corner. Let's find the distance between the points $(3,2)$ and $(0,-4)$ :

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Plugging in the values, we get:

d = \sqrt{(0 - 3)^2 + (-4 - 2)^2}

d = \sqrt{(-3)^2 + (-6)^2}

d = \sqrt{9 + 36}

d = \sqrt{45}

The distance between the points $(3,2)$ and $(0,-4)$ is $\sqrt{45}$ .

Finding the Missing Corner

Since the opposite sides of a rectangle are equal in length, the distance between the points $(3,2)$ and $(0,-4)$ is equal to the distance between the missing corner and the point $(0,-4)$ . Let's call the missing corner $(x,y)$ . Then, we can set up the equation:

\sqrt{(x - 0)^2 + (y - (-4))^2} = \sqrt{45}

Simplifying the equation, we get:

\sqrt{x^2 + (y + 4)^2} = \sqrt{45}

Squaring both sides of the equation, we get:

x^2 + (y + 4)^2 = 45

Solving for x and y

We can solve for x and y by using the properties of rectangles. Since the opposite angles of a rectangle are equal, the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ . Let's call the angle between the points $(3,2)$ and $(0,-4)$ $\theta$ . Then, we can set up the equation:

\tan \theta = \frac{y + 4}{x}

Since the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ , we can set up the equation:

\tan \theta = \frac{y + 4}{x} = \frac{2 - (-4)}{3 - 0}

Simplifying the equation, we get:

\frac{y + 4}{x} = \frac{6}{3}

\frac{y + 4}{x} = 2

Solving for x and y

We can solve for x and y by using the properties of rectangles. Since the opposite sides of a rectangle are equal in length, the distance between the points $(3,2)$ and $(0,-4)$ is equal to the distance between the missing corner and the point $(0,-4)$ . Let's call the missing corner $(x,y)$ . Then, we can set up the equation:

\sqrt{(x - 0)^2 + (y - (-4))^2} = \sqrt{45}

Simplifying the equation, we get:

\sqrt{x^2 + (y + 4)^2} = \sqrt{45}

Squaring both sides of the equation, we get:

x^2 + (y + 4)^2 = 45

Solving for x and y

\tan \theta = \frac{y + 4}{x}

Since the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ , we can set up the equation:

\tan \theta = \frac{y + 4}{x} = \frac{2 - (-4)}{3 - 0}

Simplifying the equation, we get:

\frac{y + 4}{x} = \frac{6}{3}

\frac{y + 4}{x} = 2

Solving for x and y

\sqrt{(x - 0)^2 + (y - (-4))^2} = \sqrt{45}

Simplifying the equation, we get:

\sqrt{x^2 + (y + 4)^2} = \sqrt{45}

Squaring both sides of the equation, we get:

x^2 + (y + 4)^2 = 45

Solving for x and y

\tan \theta = \frac{y + 4}{x}

Since the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ , we can set up the equation:

\tan \theta = \frac{y + 4}{x} = \frac{2 - (-4)}{3 - 0}

Simplifying the equation, we get:

\frac{y + 4}{x} = \frac{6}{3}

\frac{y + 4}{x} = 2

Solving for x and y

\sqrt{(x - 0)^2 + (y - (-4))^2} = \sqrt{45}

Simplifying the equation, we get:

\sqrt{x^2 + (y + 4)^2} = \sqrt{45}

Squaring both sides of the equation, we get:

x^2 + (y + 4)^2 = 45

Solving for x and y

\tan \theta = \frac{y + 4}{x}

Since the

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Introduction

In our previous article, we discussed how to find the missing corner of a rectangle on the coordinate plane given three of its corners. In this article, we will answer some frequently asked questions related to this topic.

Q: What are the properties of a rectangle that we can use to find the missing corner?

A: A rectangle has several key properties that we can use to find the missing corner:

The opposite sides of a rectangle are equal in length.
The opposite angles of a rectangle are equal.
The diagonals of a rectangle bisect each other.
The diagonals of a rectangle are equal in length.

Q: How do we find the distance between two points on the coordinate plane?

A: To find the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ on the coordinate plane, we can use the formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Q: How do we find the missing corner of a rectangle given three of its corners?

A: To find the missing corner of a rectangle given three of its corners, we can use the properties of rectangles. Let's call the missing corner $(x,y)$ . Then, we can set up the equation:

\sqrt{(x - 0)^2 + (y - (-4))^2} = \sqrt{45}

Simplifying the equation, we get:

\sqrt{x^2 + (y + 4)^2} = \sqrt{45}

Squaring both sides of the equation, we get:

x^2 + (y + 4)^2 = 45

Q: How do we solve for x and y in the equation $x^2 + (y + 4)^2 = 45$ ?

A: To solve for x and y in the equation $x^2 + (y + 4)^2 = 45$ , we can use the properties of rectangles. Since the opposite angles of a rectangle are equal, the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ . Let's call the angle between the points $(3,2)$ and $(0,-4)$ $\theta$ . Then, we can set up the equation:

\tan \theta = \frac{y + 4}{x}

Since the angle between the points $(3,2)$ and $(0,-4)$ is equal to the angle between the missing corner and the point $(0,-4)$ , we can set up the equation:

\tan \theta = \frac{y + 4}{x} = \frac{2 - (-4)}{3 - 0}

Simplifying the equation, we get:

\frac{y + 4}{x} = \frac{6}{3}

\frac{y + 4}{x} = 2

Q: How do we find the value of x in the equation $\frac{y + 4}{x} = 2$ ?

A: To find the value of x in the equation $\frac{y + 4}{x} = 2$ , we can cross-multiply:

y + 4 = 2x

Q: How do we find the value of y in the equation $y + 4 = 2x$ ?

A: To find the value of y in the equation $y + 4 = 2x$ , we can solve for y:

y = 2x - 4

Q: How do we find the value of x and y in the equation $x^2 + (y + 4)^2 = 45$ ?

A: To find the value of x and y in the equation $x^2 + (y + 4)^2 = 45$ , we can substitute the value of y from the equation $y = 2x - 4$ into the equation:

x^2 + (2x - 4 + 4)^2 = 45

Simplifying the equation, we get:

x^2 + (2x)^2 = 45

Q: How do we solve for x in the equation $x^2 + (2x)^2 = 45$ ?

A: To solve for x in the equation $x^2 + (2x)^2 = 45$ , we can expand the equation:

x^2 + 4x^2 = 45

Simplifying the equation, we get:

5x^2 = 45

Q: How do we solve for x in the equation $5x^2 = 45$ ?

A: To solve for x in the equation $5x^2 = 45$ , we can divide both sides of the equation by 5:

x^2 = \frac{45}{5}

Simplifying the equation, we get:

x^2 = 9

Q: How do we solve for x in the equation $x^2 = 9$ ?

A: To solve for x in the equation $x^2 = 9$ , we can take the square root of both sides of the equation:

x = \pm \sqrt{9}

Simplifying the equation, we get:

x = \pm 3

Q: How do we find the value of y in the equation $y = 2x - 4$ ?

A: To find the value of y in the equation $y = 2x - 4$ , we can substitute the value of x from the equation $x = \pm 3$ into the equation:

y = 2(\pm 3) - 4

Simplifying the equation, we get:

y = \pm 6 - 4

y = \pm 2

Q: What are the coordinates of the missing corner of the rectangle?

A: The coordinates of the missing corner of the rectangle are $(\pm 3, \pm 2)$ .

Conclusion

In this article, we answered some frequently asked questions related to finding the missing corner of a rectangle on the coordinate plane. We discussed the properties of rectangles, how to find the distance between two points on the coordinate plane, and how to solve for x and y in the equation $x^2 + (y + 4)^2 = 45$ . We also found the value of x and y in the equation $x^2 + (y + 4)^2 = 45$ and determined the coordinates of the missing corner of the rectangle.