Graph A Right Triangle With The Two Points Forming The Hypotenuse. Using The Sides, Find The Distance Between The Two Points, To The Nearest Tenth (if Necessary).Points: \[$(9,6)\$\] And \[$(2,-1)\$\]Leg 1: \[$\square\$\] Leg

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Introduction

In this article, we will explore how to graph a right triangle using two points that form the hypotenuse. We will also learn how to find the distance between these two points using the sides of the triangle. This is a fundamental concept in mathematics, particularly in geometry and trigonometry.

Graphing the Right Triangle

To graph a right triangle, we need to identify the two points that form the hypotenuse. In this case, the two points are (9, 6) and (2, -1). We will use these points to graph the triangle.

Step 1: Plot the Points

First, we need to plot the two points on a coordinate plane. To do this, we will use the x-coordinate and y-coordinate of each point to determine its location on the plane.

  • Point 1: (9, 6)
  • Point 2: (2, -1)

Step 2: Draw the Legs of the Triangle

Next, we need to draw the legs of the triangle. The legs are the two sides of the triangle that meet at a right angle. In this case, the legs are the line segments connecting the two points.

  • Leg 1: Connect point 1 (9, 6) to point 2 (2, -1)
  • Leg 2: Connect point 1 (9, 6) to the origin (0, 0)

Step 3: Draw the Hypotenuse

The hypotenuse is the longest side of the triangle, opposite the right angle. In this case, the hypotenuse is the line segment connecting the two points.

  • Hypotenuse: Connect point 1 (9, 6) to point 2 (2, -1)

Finding the Distance Between the Two Points

Now that we have graphed the right triangle, we can use the sides of the triangle to find the distance between the two points. We will use the Pythagorean theorem to calculate the distance.

The Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

a^2 + b^2 = c^2

Step 1: Identify the Sides of the Triangle

To use the Pythagorean theorem, we need to identify the sides of the triangle. In this case, the sides are:

  • Leg 1: The line segment connecting point 1 (9, 6) to the origin (0, 0)
  • Leg 2: The line segment connecting point 1 (9, 6) to point 2 (2, -1)
  • Hypotenuse: The line segment connecting point 1 (9, 6) to point 2 (2, -1)

Step 2: Calculate the Lengths of the Sides

To calculate the lengths of the sides, we will use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Step 3: Calculate the Length of Leg 1

To calculate the length of leg 1, we will use the distance formula.

  • Leg 1: Connect point 1 (9, 6) to the origin (0, 0)
  • x1 = 9
  • y1 = 6
  • x2 = 0
  • y2 = 0

d = √((x2 - x1)^2 + (y2 - y1)^2) d = √((0 - 9)^2 + (0 - 6)^2) d = √(81 + 36) d = √117 d ≈ 10.8

Step 4: Calculate the Length of Leg 2

To calculate the length of leg 2, we will use the distance formula.

  • Leg 2: Connect point 1 (9, 6) to point 2 (2, -1)
  • x1 = 9
  • y1 = 6
  • x2 = 2
  • y2 = -1

d = √((x2 - x1)^2 + (y2 - y1)^2) d = √((2 - 9)^2 + (-1 - 6)^2) d = √(49 + 49) d = √98 d ≈ 9.9

Step 5: Calculate the Length of the Hypotenuse

To calculate the length of the hypotenuse, we will use the Pythagorean theorem.

  • a = 10.8 (length of leg 1)
  • b = 9.9 (length of leg 2)
  • c = ? (length of the hypotenuse)

a^2 + b^2 = c^2 (10.8)^2 + (9.9)^2 = c^2 116.64 + 98.01 = c^2 214.65 = c^2 c = √214.65 c ≈ 14.7

Conclusion

Introduction

In our previous article, we explored how to graph a right triangle using two points that form the hypotenuse. We also learned how to find the distance between these two points using the sides of the triangle. In this article, we will answer some frequently asked questions about graphing a right triangle and finding the distance between two points.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

a^2 + b^2 = c^2

Q: How do I graph a right triangle?

A: To graph a right triangle, you need to identify the two points that form the hypotenuse. You can then use these points to graph the triangle. To do this, follow these steps:

  1. Plot the two points on a coordinate plane.
  2. Draw the legs of the triangle by connecting the two points.
  3. Draw the hypotenuse by connecting the two points.

Q: How do I find the distance between two points?

A: To find the distance between two points, you can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Q: What is the difference between the distance formula and the Pythagorean theorem?

A: The distance formula and the Pythagorean theorem are both used to find the distance between two points. However, the distance formula is used to find the distance between two points in a coordinate plane, while the Pythagorean theorem is used to find the length of the hypotenuse of a right triangle.

Q: Can I use the Pythagorean theorem to find the distance between two points?

A: Yes, you can use the Pythagorean theorem to find the distance between two points. However, you need to first find the lengths of the legs of the triangle using the distance formula.

Q: What is the significance of the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in mathematics, particularly in geometry and trigonometry. It is used to find the length of the hypotenuse of a right triangle, which is essential in many real-world applications, such as architecture, engineering, and physics.

Q: Can I use the Pythagorean theorem to find the length of a side of a triangle that is not a right triangle?

A: No, you cannot use the Pythagorean theorem to find the length of a side of a triangle that is not a right triangle. The Pythagorean theorem only applies to right triangles.

Q: What are some real-world applications of the Pythagorean theorem?

A: The Pythagorean theorem has many real-world applications, such as:

  • Architecture: The Pythagorean theorem is used to design buildings and bridges.
  • Engineering: The Pythagorean theorem is used to design machines and mechanisms.
  • Physics: The Pythagorean theorem is used to describe the motion of objects.
  • Navigation: The Pythagorean theorem is used to calculate distances and directions.

Conclusion

In this article, we answered some frequently asked questions about graphing a right triangle and finding the distance between two points. We also discussed the significance of the Pythagorean theorem and its real-world applications. We hope that this article has been helpful in understanding the concept of graphing a right triangle and finding the distance between two points.