Finding the Length of Side AB

Oct 13, 2025 by miodeya.com 105 views

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Introduction

In geometry, a triangle is a polygon with three sides and three vertices. Given the coordinates of the vertices, we can use the distance formula to find the lengths of the sides of the triangle. In this article, we will find the lengths of sides AB and AC of a triangle with points A(3,6), B(3,1), and C(7,3).

The Distance Formula

The distance formula is a mathematical formula that gives the distance between two points in a coordinate plane. The formula is:

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

To find the length of side AB, we will use the distance formula with the coordinates of points A and B.

The coordinates of point A are (3, 6).
The coordinates of point B are (3, 1).

Using the distance formula, we get:

AB = \sqrt{(3 - 3)^2 + (1 - 6)^2}

AB = \sqrt{0^2 + (-5)^2}

AB = \sqrt{0 + 25}

AB = \sqrt{25}

AB = 5

Therefore, the length of side AB is 5 units.

Finding the Length of Side AC

To find the length of side AC, we will use the distance formula with the coordinates of points A and C.

The coordinates of point A are (3, 6).
The coordinates of point C are (7, 3).

Using the distance formula, we get:

AC = \sqrt{(7 - 3)^2 + (3 - 6)^2}

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

AC = \sqrt{16 + 9}

AC = \sqrt{25}

AC = 5

Therefore, the length of side AC is 5 units.

Conclusion

In this article, we found the lengths of sides AB and AC of a triangle with points A(3,6), B(3,1), and C(7,3). We used the distance formula to find the lengths of the sides. The length of side AB is 5 units, and the length of side AC is also 5 units.

Discussion

The distance formula is a powerful tool in geometry that allows us to find the distance between two points in a coordinate plane. It is a fundamental concept in mathematics and has many applications in real-life situations.

Example Use Cases

The distance formula has many applications in real-life situations. Here are a few examples:

GPS Navigation: The distance formula is used in GPS navigation systems to calculate the distance between two points on the Earth's surface.
Surveying: The distance formula is used in surveying to calculate the distance between two points on a map.
Physics: The distance formula is used in physics to calculate the distance traveled by an object.

Exercises

Here are a few exercises to practice finding the lengths of sides of a triangle using the distance formula:

Find the length of side AB of a triangle with points A(2, 4), B(2, 1), and C(6, 3).
Find the length of side AC of a triangle with points A(1, 2), B(4, 2), and C(4, 5).
Find the length of side BC of a triangle with points A(3, 6), B(3, 1), and C(7, 3).

Solutions

Here are the solutions to the exercises:

The length of side AB is $\sqrt{(2 - 2)^2 + (1 - 4)^2} = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3$ units.
The length of side AC is $\sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$ units.
The length of side BC is $\sqrt{(7 - 3)^2 + (3 - 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$ units.

Conclusion

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Introduction

In our previous article, we found the lengths of sides AB and AC of a triangle with points A(3,6), B(3,1), and C(7,3). We used the distance formula to find the lengths of the sides. In this article, we will answer some frequently asked questions about the distance formula and its applications.

Q&A

Q: What is the distance formula?

A: The distance formula is a mathematical formula that gives the distance between two points in a coordinate plane. The formula is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^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 use the distance formula to find the length of a side of a triangle?

A: To use the distance formula to find the length of a side of a triangle, you need to know the coordinates of the two points that define the side. You can then plug these coordinates into the distance formula and solve for the distance.

Q: What are some real-life applications of the distance formula?

A: The distance formula has many real-life applications, including:

GPS Navigation: The distance formula is used in GPS navigation systems to calculate the distance between two points on the Earth's surface.
Surveying: The distance formula is used in surveying to calculate the distance between two points on a map.
Physics: The distance formula is used in physics to calculate the distance traveled by an object.

Q: Can I use the distance formula to find the length of a diagonal of a rectangle?

A: Yes, you can use the distance formula to find the length of a diagonal of a rectangle. To do this, you need to know the coordinates of the two opposite corners of the rectangle. You can then plug these coordinates into the distance formula and solve for the distance.

Q: How do I find the length of a side of a triangle if I only know the coordinates of the vertices?

A: To find the length of a side of a triangle if you only know the coordinates of the vertices, you can use the distance formula. You need to know the coordinates of the two points that define the side, and you can then plug these coordinates into the distance formula and solve for the distance.

Q: Can I use the distance formula to find the length of a side of a triangle if the triangle is not a right triangle?

A: Yes, you can use the distance formula to find the length of a side of a triangle even if the triangle is not a right triangle. The distance formula works for all types of triangles.

Conclusion

In this article, we answered some frequently asked questions about the distance formula and its applications. We discussed how to use the distance formula to find the length of a side of a triangle, and we talked about some real-life applications of the distance formula. We also answered questions about using the distance formula to find the length of a diagonal of a rectangle and to find the length of a side of a triangle if the triangle is not a right triangle.

Discussion

Example Use Cases

The distance formula has many applications in real-life situations. Here are a few examples:

GPS Navigation: The distance formula is used in GPS navigation systems to calculate the distance between two points on the Earth's surface.
Surveying: The distance formula is used in surveying to calculate the distance between two points on a map.
Physics: The distance formula is used in physics to calculate the distance traveled by an object.

Exercises

Here are a few exercises to practice using the distance formula to find the length of a side of a triangle:

Find the length of side AB of a triangle with points A(2, 4), B(2, 1), and C(6, 3).
Find the length of side AC of a triangle with points A(1, 2), B(4, 2), and C(4, 5).
Find the length of side BC of a triangle with points A(3, 6), B(3, 1), and C(7, 3).

Solutions

Here are the solutions to the exercises:

The length of side AB is $\sqrt{(2 - 2)^2 + (1 - 4)^2} = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3$ units.
The length of side AC is $\sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$ units.
The length of side BC is $\sqrt{(7 - 3)^2 + (3 - 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$ units.