On A Coordinate Plane, Triangle X Y Z Is Shown. Point X Is At (1, 3), Point Y Is At (4, Negative 1), And Point Z Is At (5, 6). Which Statement Proves That △XYZ Is An Isosceles Right Triangle?

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Introduction

In geometry, an isosceles right triangle is a special type of triangle that has two sides of equal length and one right angle. When dealing with triangles on a coordinate plane, it can be challenging to determine if a triangle is an isosceles right triangle. In this article, we will explore how to identify an isosceles right triangle on a coordinate plane and provide a statement that proves △XYZ is an isosceles right triangle.

What is an Isosceles Right Triangle?

An isosceles right triangle is a triangle with two sides of equal length and one right angle. The two sides of equal length are called the legs, and the side opposite the right angle is called the hypotenuse. In an isosceles right triangle, the legs are equal in length, and the hypotenuse is equal to the square root of 2 times the length of a leg.

Properties of Isosceles Right Triangles

Isosceles right triangles have several properties that can be used to identify them on a coordinate plane. Some of these properties include:

  • The legs of the triangle are equal in length.
  • The hypotenuse is equal to the square root of 2 times the length of a leg.
  • The triangle has one right angle.
  • The triangle is symmetrical about the perpendicular bisector of the hypotenuse.

Identifying Isosceles Right Triangles on a Coordinate Plane

To identify an isosceles right triangle on a coordinate plane, we need to use the distance formula to calculate the lengths of the sides of the triangle. The distance formula is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between two points (x1, y1) and (x2, y2).

Example: Identifying △XYZ as an Isosceles Right Triangle

Let's use the points X (1, 3), Y (4, -1), and Z (5, 6) to identify △XYZ as an isosceles right triangle.

Step 1: Calculate the Lengths of the Sides of △XYZ

To calculate the lengths of the sides of △XYZ, we need to use the distance formula.

  • The distance between X (1, 3) and Y (4, -1) is:

d1 = √((4 - 1)^2 + (-1 - 3)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5

  • The distance between Y (4, -1) and Z (5, 6) is:

d2 = √((5 - 4)^2 + (6 - (-1))^2) = √(1^2 + 7^2) = √(1 + 49) = √50

  • The distance between X (1, 3) and Z (5, 6) is:

d3 = √((5 - 1)^2 + (6 - 3)^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5

Step 2: Determine if △XYZ is an Isosceles Right Triangle

From the calculations above, we can see that the lengths of the sides of △XYZ are:

  • d1 = 5
  • d2 = √50
  • d3 = 5

Since d1 = d3, we know that △XYZ is an isosceles triangle. Additionally, since d2^2 = d1^2 + d3^2, we know that △XYZ is a right triangle.

Conclusion

In conclusion, we have identified △XYZ as an isosceles right triangle using the distance formula and the properties of isosceles right triangles. The statement that proves △XYZ is an isosceles right triangle is:

"The triangle △XYZ is an isosceles right triangle because it has two sides of equal length (d1 = d3 = 5) and one right angle (d2^2 = d1^2 + d3^2)."

Properties of Isosceles Right Triangles on a Coordinate Plane

Isosceles right triangles on a coordinate plane have several properties that can be used to identify them. Some of these properties include:

  • The legs of the triangle are equal in length.
  • The hypotenuse is equal to the square root of 2 times the length of a leg.
  • The triangle has one right angle.
  • The triangle is symmetrical about the perpendicular bisector of the hypotenuse.

Example: Identifying Isosceles Right Triangles on a Coordinate Plane

Let's use the points A (2, 2), B (4, 4), and C (6, 6) to identify △ABC as an isosceles right triangle.

Step 1: Calculate the Lengths of the Sides of △ABC

To calculate the lengths of the sides of △ABC, we need to use the distance formula.

  • The distance between A (2, 2) and B (4, 4) is:

d1 = √((4 - 2)^2 + (4 - 2)^2) = √(2^2 + 2^2) = √(4 + 4) = √8

  • The distance between B (4, 4) and C (6, 6) is:

d2 = √((6 - 4)^2 + (6 - 4)^2) = √(2^2 + 2^2) = √(4 + 4) = √8

  • The distance between A (2, 2) and C (6, 6) is:

d3 = √((6 - 2)^2 + (6 - 2)^2) = √(4^2 + 4^2) = √(16 + 16) = √32

Step 2: Determine if △ABC is an Isosceles Right Triangle

From the calculations above, we can see that the lengths of the sides of △ABC are:

  • d1 = √8
  • d2 = √8
  • d3 = √32

Since d1 = d2, we know that △ABC is an isosceles triangle. Additionally, since d3^2 = 2d1^2, we know that △ABC is a right triangle.

Conclusion

In conclusion, we have identified △ABC as an isosceles right triangle using the distance formula and the properties of isosceles right triangles. The statement that proves △ABC is an isosceles right triangle is:

"The triangle △ABC is an isosceles right triangle because it has two sides of equal length (d1 = d2 = √8) and one right angle (d3^2 = 2d1^2)."

Properties of Isosceles Right Triangles in Real-World Applications

Isosceles right triangles have several real-world applications, including:

  • Architecture: Isosceles right triangles are used in the design of buildings and bridges to create symmetrical and aesthetically pleasing structures.
  • Engineering: Isosceles right triangles are used in the design of machines and mechanisms to create efficient and reliable systems.
  • Art: Isosceles right triangles are used in the creation of art and design to create symmetrical and balanced compositions.

Conclusion

Q: What is an isosceles right triangle?

A: An isosceles right triangle is a special type of triangle that has two sides of equal length and one right angle. The two sides of equal length are called the legs, and the side opposite the right angle is called the hypotenuse.

Q: How do I identify an isosceles right triangle on a coordinate plane?

A: To identify an isosceles right triangle on a coordinate plane, you need to use the distance formula to calculate the lengths of the sides of the triangle. If the lengths of the two legs are equal, and the length of the hypotenuse is equal to the square root of 2 times the length of a leg, then the triangle is an isosceles right triangle.

Q: What are the properties of isosceles right triangles?

A: The properties of isosceles right triangles include:

  • The legs of the triangle are equal in length.
  • The hypotenuse is equal to the square root of 2 times the length of a leg.
  • The triangle has one right angle.
  • The triangle is symmetrical about the perpendicular bisector of the hypotenuse.

Q: How do I calculate the length of the hypotenuse of an isosceles right triangle?

A: To calculate the length of the hypotenuse of an isosceles right triangle, you can use the formula:

c = √2a

where c is the length of the hypotenuse, and a is the length of one of the legs.

Q: What are some real-world applications of isosceles right triangles?

A: Isosceles right triangles have several real-world applications, including:

  • Architecture: Isosceles right triangles are used in the design of buildings and bridges to create symmetrical and aesthetically pleasing structures.
  • Engineering: Isosceles right triangles are used in the design of machines and mechanisms to create efficient and reliable systems.
  • Art: Isosceles right triangles are used in the creation of art and design to create symmetrical and balanced compositions.

Q: How do I use isosceles right triangles in art and design?

A: Isosceles right triangles can be used in art and design to create symmetrical and balanced compositions. You can use isosceles right triangles to create patterns, shapes, and designs that are aesthetically pleasing and visually appealing.

Q: What are some common mistakes to avoid when working with isosceles right triangles?

A: Some common mistakes to avoid when working with isosceles right triangles include:

  • Not checking for equality of the legs: Make sure to check that the lengths of the two legs are equal before concluding that a triangle is an isosceles right triangle.
  • Not checking for the right angle: Make sure to check that the triangle has one right angle before concluding that it is an isosceles right triangle.
  • Not using the correct formula for the hypotenuse: Make sure to use the correct formula for the hypotenuse, which is c = √2a.

Q: How do I practice working with isosceles right triangles?

A: You can practice working with isosceles right triangles by:

  • Drawing and sketching: Draw and sketch isosceles right triangles to practice identifying and creating them.
  • Calculating lengths: Calculate the lengths of the sides of isosceles right triangles to practice using the distance formula and the properties of isosceles right triangles.
  • Solving problems: Solve problems that involve isosceles right triangles to practice applying the properties and formulas of isosceles right triangles.

Conclusion

In conclusion, isosceles right triangles are an important concept in geometry and have several real-world applications. By understanding the properties and formulas of isosceles right triangles, you can identify and create them on a coordinate plane and use them to create symmetrical and aesthetically pleasing structures.