Use The Drop-downs To Answer The Following Questions About The Distance Between The Points $(-5, 1$\] And $(2, -1$\].1. What Is The Distance Of The Horizontal Leg? $\square$2. What Is The Distance Of The Vertical Leg?
Introduction
In mathematics, the distance between two points is a fundamental concept that has numerous applications in various fields, including geometry, trigonometry, and physics. The distance between two points can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides. In this article, we will explore how to calculate the distance between two points using the Pythagorean theorem and discuss the horizontal and vertical legs of the triangle.
Understanding the Problem
The problem asks us to find the distance between two points, (-5, 1) and (2, -1). To solve this problem, we need to understand the concept of distance and how to calculate it using the Pythagorean theorem.
Calculating the Distance
The distance between two points can be calculated using the following formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Step 1: Identify the Coordinates
The coordinates of the two points are (-5, 1) and (2, -1).
Step 2: Plug in the Values
We can now plug in the values into the formula:
d = √((2 - (-5))^2 + (-1 - 1)^2)
Step 3: Simplify the Expression
We can simplify the expression by evaluating the expressions inside the parentheses:
d = √((7)^2 + (-2)^2)
Step 4: Calculate the Squares
We can now calculate the squares:
d = √(49 + 4)
Step 5: Add the Numbers
We can now add the numbers:
d = √53
Step 6: Calculate the Square Root
We can now calculate the square root:
d ≈ 7.28
The Horizontal Leg
The horizontal leg of the triangle is the distance between the x-coordinates of the two points. To calculate the horizontal leg, we can use the following formula:
horizontal leg = |x2 - x1|
where x1 and x2 are the x-coordinates of the two points.
Step 1: Identify the X-Coordinates
The x-coordinates of the two points are -5 and 2.
Step 2: Plug in the Values
We can now plug in the values into the formula:
horizontal leg = |2 - (-5)|
Step 3: Simplify the Expression
We can simplify the expression by evaluating the expression inside the absolute value:
horizontal leg = |7|
Step 4: Calculate the Absolute Value
We can now calculate the absolute value:
horizontal leg = 7
The Vertical Leg
The vertical leg of the triangle is the distance between the y-coordinates of the two points. To calculate the vertical leg, we can use the following formula:
vertical leg = |y2 - y1|
where y1 and y2 are the y-coordinates of the two points.
Step 1: Identify the Y-Coordinates
The y-coordinates of the two points are 1 and -1.
Step 2: Plug in the Values
We can now plug in the values into the formula:
vertical leg = |-1 - 1|
Step 3: Simplify the Expression
We can simplify the expression by evaluating the expression inside the absolute value:
vertical leg = |-2|
Step 4: Calculate the Absolute Value
We can now calculate the absolute value:
vertical leg = 2
Conclusion
In this article, we have discussed how to calculate the distance between two points using the Pythagorean theorem. We have also discussed the horizontal and vertical legs of the triangle and how to calculate them. The distance between two points is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to calculate the distance between two points, we can solve a wide range of problems in mathematics and other fields.
Frequently Asked Questions
Q: What is the distance between two points?
A: The distance between two points can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
Q: How do I calculate the horizontal leg of the triangle?
A: To calculate the horizontal leg of the triangle, you can use the following formula: horizontal leg = |x2 - x1|, where x1 and x2 are the x-coordinates of the two points.
Q: How do I calculate the vertical leg of the triangle?
A: To calculate the vertical leg of the triangle, you can use the following formula: vertical leg = |y2 - y1|, where y1 and y2 are the y-coordinates of the two points.
Q: What is the distance between the points (-5, 1) and (2, -1)?
A: The distance between the points (-5, 1) and (2, -1) is approximately 7.28.
Q: What is the horizontal leg of the triangle between the points (-5, 1) and (2, -1)?
A: The horizontal leg of the triangle between the points (-5, 1) and (2, -1) is 7.
Q: What is the vertical leg of the triangle between the points (-5, 1) and (2, -1)?
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a fundamental concept in mathematics that states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides. The formula for the Pythagorean theorem is: a^2 + b^2 = c^2, where a and b are the lengths of the two sides and c is the length of the hypotenuse.
Q: How do I calculate the distance between two points?
A: To calculate the distance between two points, you can use the following formula: d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the horizontal leg of the triangle?
A: The horizontal leg of the triangle is the distance between the x-coordinates of the two points. To calculate the horizontal leg, you can use the following formula: horizontal leg = |x2 - x1|, where x1 and x2 are the x-coordinates of the two points.
Q: What is the vertical leg of the triangle?
A: The vertical leg of the triangle is the distance between the y-coordinates of the two points. To calculate the vertical leg, you can use the following formula: vertical leg = |y2 - y1|, where y1 and y2 are the y-coordinates of the two points.
Q: How do I calculate the distance between two points in a 3D space?
A: To calculate the distance between two points in a 3D space, you can use the following formula: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2), where d is the distance between the two points, (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.
Q: What is the formula for the distance between two points in a 3D space?
A: The formula for the distance between two points in a 3D space is: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2), where d is the distance between the two points, (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.
Q: How do I calculate the distance between two points in a 2D space?
A: To calculate the distance between two points in a 2D space, you can use the following formula: d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the formula for the distance between two points in a 2D space?
A: The formula for the distance between two points in a 2D space is: d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: How do I calculate the distance between two points in a polar coordinate system?
A: To calculate the distance between two points in a polar coordinate system, you can use the following formula: d = √(r1^2 + r2^2 - 2r1r2cos(θ)), where d is the distance between the two points, r1 and r2 are the radii of the two points, and θ is the angle between the two points.
Q: What is the formula for the distance between two points in a polar coordinate system?
A: The formula for the distance between two points in a polar coordinate system is: d = √(r1^2 + r2^2 - 2r1r2cos(θ)), where d is the distance between the two points, r1 and r2 are the radii of the two points, and θ is the angle between the two points.
Q: How do I calculate the distance between two points in a spherical coordinate system?
A: To calculate the distance between two points in a spherical coordinate system, you can use the following formula: d = √(r1^2 + r2^2 - 2r1r2cos(θ)), where d is the distance between the two points, r1 and r2 are the radii of the two points, and θ is the angle between the two points.
Q: What is the formula for the distance between two points in a spherical coordinate system?
A: The formula for the distance between two points in a spherical coordinate system is: d = √(r1^2 + r2^2 - 2r1r2cos(θ)), where d is the distance between the two points, r1 and r2 are the radii of the two points, and θ is the angle between the two points.
Conclusion
In this article, we have discussed the concept of distance between two points and how to calculate it using various formulas. We have also discussed the horizontal and vertical legs of the triangle and how to calculate them. The distance between two points is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to calculate the distance between two points, we can solve a wide range of problems in mathematics and other fields.
Q: What is the distance between the points (3, 4) and (6, 8)?
A: The distance between the points (3, 4) and (6, 8) is approximately 5.29.
Q: What is the horizontal leg of the triangle between the points (3, 4) and (6, 8)?
A: The horizontal leg of the triangle between the points (3, 4) and (6, 8) is 3.
Q: What is the vertical leg of the triangle between the points (3, 4) and (6, 8)?
A: The vertical leg of the triangle between the points (3, 4) and (6, 8) is 4.
Q: How do I calculate the distance between two points in a 3D space using the distance formula?
A: To calculate the distance between two points in a 3D space using the distance formula, you can use the following formula: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2), where d is the distance between the two points, (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.
Q: What is the formula for the distance between two points in a 3D space using the distance formula?
A: The formula for the distance between two points in a 3D space using the distance formula is: d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2), where d is the distance between the two points, (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.
Q: How do I calculate the distance between two points in a 2D space using the distance formula?
A: To calculate the distance between two points in a 2D space using the distance formula, you can use the following formula: d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the formula for the distance between two points in a 2D space using the distance formula?
A: The formula for the distance between two points in a 2D space using the distance formula is: d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points.
Conclusion
In this article, we have discussed the concept of distance between two points and how to calculate it using various formulas. We have also discussed the horizontal and vertical legs of the triangle and how to calculate them. The distance between two points is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to calculate the distance between two points, we can solve a wide range of problems in mathematics and other fields.