An Equilateral Triangle On The Coordinate Plane Has Vertices At \[$(-1,2)\$\] And \[$(4,2)\$\]. Select All The Possible Coordinates Of The Third Vertex, Rounded To The Nearest Tenth:A. \[$(-1.5, 6.3)\$\] B. \[$(1.5,
Introduction
In geometry, an equilateral triangle is a triangle with all sides of equal length. When dealing with equilateral triangles on the coordinate plane, we can use the distance formula to find the length of the sides and determine the possible coordinates of the third vertex. In this article, we will explore how to find the third vertex of an equilateral triangle given the coordinates of two vertices.
Understanding the Problem
We are given the coordinates of two vertices of an equilateral triangle: (-1, 2) and (4, 2). We need to find the possible coordinates of the third vertex, rounded to the nearest tenth. To do this, we will use the distance formula to find the length of the sides of the triangle and then determine the possible coordinates of the third vertex.
The Distance Formula
The distance formula is a fundamental concept in geometry that allows us to find the distance between two points on the coordinate plane. The distance formula is given by:
d = β((x2 - x1)^2 + (y2 - y1)^2)
where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Finding the Length of the Sides
To find the length of the sides of the equilateral triangle, we can use the distance formula. Let's find the distance between the two given vertices:
d = β((4 - (-1))^2 + (2 - 2)^2) d = β((5)^2 + (0)^2) d = β(25) d = 5
Since the triangle is equilateral, all sides have the same length. Therefore, the length of each side is 5 units.
Finding the Possible Coordinates of the Third Vertex
To find the possible coordinates of the third vertex, we need to consider the properties of an equilateral triangle. An equilateral triangle has three equal sides and three equal angles (each angle is 60 degrees). Since the two given vertices are on the same horizontal line (y-coordinate is the same), the third vertex must be on a line that is perpendicular to this line.
Option A: (-1.5, 6.3)
Let's consider the first option: (-1.5, 6.3). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (-1.5, 6.3) and (-1, 2): d = β((-1.5 - (-1))^2 + (6.3 - 2)^2) d = β((-0.5)^2 + (4.3)^2) d = β(0.25 + 18.49) d = β(18.74) d β 4.33
Distance between (-1.5, 6.3) and (4, 2): d = β((-1.5 - 4)^2 + (6.3 - 2)^2) d = β((-5.5)^2 + (4.3)^2) d = β(30.25 + 18.49) d = β(48.74) d β 6.98
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Option B: (1.5, 2)
Let's consider the second option: (1.5, 2). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (1.5, 2) and (-1, 2): d = β((1.5 - (-1))^2 + (2 - 2)^2) d = β((2.5)^2 + (0)^2) d = β(6.25) d β 2.5
Distance between (1.5, 2) and (4, 2): d = β((1.5 - 4)^2 + (2 - 2)^2) d = β((-2.5)^2 + (0)^2) d = β(6.25) d β 2.5
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Option C: (1.5, 6.3)
Let's consider the third option: (1.5, 6.3). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (1.5, 6.3) and (-1, 2): d = β((1.5 - (-1))^2 + (6.3 - 2)^2) d = β((2.5)^2 + (4.3)^2) d = β(6.25 + 18.49) d = β(24.74) d β 4.96
Distance between (1.5, 6.3) and (4, 2): d = β((1.5 - 4)^2 + (6.3 - 2)^2) d = β((-2.5)^2 + (4.3)^2) d = β(6.25 + 18.49) d = β(24.74) d β 4.96
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Option D: (-1.5, 2)
Let's consider the fourth option: (-1.5, 2). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (-1.5, 2) and (-1, 2): d = β((-1.5 - (-1))^2 + (2 - 2)^2) d = β((-0.5)^2 + (0)^2) d = β(0.25) d β 0.5
Distance between (-1.5, 2) and (4, 2): d = β((-1.5 - 4)^2 + (2 - 2)^2) d = β((-5.5)^2 + (0)^2) d = β(30.25) d β 5.5
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Option E: (-1.5, 2.5)
Let's consider the fifth option: (-1.5, 2.5). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (-1.5, 2.5) and (-1, 2): d = β((-1.5 - (-1))^2 + (2.5 - 2)^2) d = β((-0.5)^2 + (0.5)^2) d = β(0.25 + 0.25) d = β(0.5) d β 0.71
Distance between (-1.5, 2.5) and (4, 2): d = β((-1.5 - 4)^2 + (2.5 - 2)^2) d = β((-5.5)^2 + (0.5)^2) d = β(30.25 + 0.25) d = β(30.5) d β 5.52
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Option F: (-1.5, 2.5)
Let's consider the sixth option: (-1.5, 2.5). To determine if this is a possible coordinate of the third vertex, we need to check if the distance between this point and the two given vertices is equal to the length of the side (5 units).
Distance between (-1.5, 2.5) and (-1, 2): d = β((-1.5 - (-1))^2 + (2.5 - 2)^2) d = β((-0.5)^2 + (0.5)^2) d = β(0.25 + 0.25) d = β(0.5) d β 0.71
Distance between (-1.5, 2.5) and (4, 2): d = β((-1.5 - 4)^2 + (2.5 - 2)^2) d = β((-5.5)^2 + (0.5)^2) d = β(30.25 + 0.25) d = β(30.5) d β 5.52
Since the distances are not equal to 5 units, this option is not a possible coordinate of the third vertex.
Conclusion
After analyzing the options, we can conclude that none of the options A, B, C, D, E, or F are possible coordinates of the third vertex. However, we can find the correct option by using the properties of an equilateral triangle.
Finding the Correct Option
To find the correct
Introduction
In our previous article, we explored how to find the third vertex of an equilateral triangle given the coordinates of two vertices. We analyzed several options and concluded that none of them were correct. In this article, we will provide a Q&A section to help clarify any doubts and provide a step-by-step guide to finding the correct option.
Q: What is an equilateral triangle?
A: An equilateral triangle is a triangle with all sides of equal length. In other words, all three sides of an equilateral triangle are congruent.
Q: How do I find the length of the sides of an equilateral triangle?
A: To find the length of the sides of an equilateral triangle, you can use the distance formula. The distance formula is given by:
d = β((x2 - x1)^2 + (y2 - y1)^2)
where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: How do I find the possible coordinates of the third vertex?
A: To find the possible coordinates of the third vertex, you need to consider the properties of an equilateral triangle. An equilateral triangle has three equal sides and three equal angles (each angle is 60 degrees). Since the two given vertices are on the same horizontal line (y-coordinate is the same), the third vertex must be on a line that is perpendicular to this line.
Q: What is the correct option for the third vertex?
A: To find the correct option, you need to use the properties of an equilateral triangle. Since the two given vertices are on the same horizontal line (y-coordinate is the same), the third vertex must be on a line that is perpendicular to this line. The correct option is the one that satisfies this condition.
Q: How do I determine if a point is on a line that is perpendicular to a given line?
A: To determine if a point is on a line that is perpendicular to a given line, you need to check if the slope of the line is the negative reciprocal of the slope of the given line. The slope of a line is given by:
m = (y2 - y1) / (x2 - x1)
where m is the slope of the line, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the slope of the line that passes through the two given vertices?
A: The slope of the line that passes through the two given vertices is given by:
m = (2 - 2) / (4 - (-1)) m = 0 / 5 m = 0
Q: What is the slope of the line that is perpendicular to the line that passes through the two given vertices?
A: The slope of the line that is perpendicular to the line that passes through the two given vertices is the negative reciprocal of the slope of the line that passes through the two given vertices. Since the slope of the line that passes through the two given vertices is 0, the slope of the line that is perpendicular to the line that passes through the two given vertices is undefined.
Q: What does it mean for the slope of a line to be undefined?
A: When the slope of a line is undefined, it means that the line is vertical. A vertical line has an undefined slope because the denominator of the slope formula is zero.
Q: What is the equation of the line that is perpendicular to the line that passes through the two given vertices?
A: Since the line that is perpendicular to the line that passes through the two given vertices is vertical, its equation is x = -1.
Q: What are the possible coordinates of the third vertex?
A: The possible coordinates of the third vertex are the points that lie on the line x = -1 and satisfy the condition that the distance between the point and the two given vertices is equal to the length of the side (5 units).
Q: How do I find the possible coordinates of the third vertex?
A: To find the possible coordinates of the third vertex, you need to use the equation of the line that is perpendicular to the line that passes through the two given vertices (x = -1) and the condition that the distance between the point and the two given vertices is equal to the length of the side (5 units).
Q: What are the possible coordinates of the third vertex?
A: After analyzing the options, we can conclude that the possible coordinates of the third vertex are (-1.5, 6.3) and (1.5, 2).
Q: Why are these options the possible coordinates of the third vertex?
A: These options are the possible coordinates of the third vertex because they satisfy the condition that the distance between the point and the two given vertices is equal to the length of the side (5 units) and lie on the line x = -1.
Q: How do I determine which of the two options is the correct option?
A: To determine which of the two options is the correct option, you need to use the properties of an equilateral triangle. Since the two given vertices are on the same horizontal line (y-coordinate is the same), the third vertex must be on a line that is perpendicular to this line. The correct option is the one that satisfies this condition.
Q: Which option satisfies this condition?
A: The option (-1.5, 6.3) satisfies this condition because it lies on the line x = -1 and has a y-coordinate that is greater than the y-coordinate of the two given vertices.
Q: Why is this option the correct option?
A: This option is the correct option because it satisfies the condition that the distance between the point and the two given vertices is equal to the length of the side (5 units) and lies on the line x = -1.
Conclusion
In this Q&A article, we provided a step-by-step guide to finding the possible coordinates of the third vertex of an equilateral triangle given the coordinates of two vertices. We also determined which of the two options is the correct option.