Understanding Isosceles Right Triangles

Oct 13, 2025 by vinooalluri.com 145 views

Which Statement Proves That $\Delta XYZ$ Is an Isosceles Right Triangle?

In geometry, an isosceles right triangle is a special type of triangle where two sides are equal in length, and the third side is the hypotenuse, which is equal to the square root of the sum of the squares of the other two sides. In this article, we will explore which statement proves that $\Delta XYZ$ is an isosceles right triangle.

Understanding Isosceles Right Triangles

An isosceles right triangle has two equal sides, which are the legs of the triangle, and the third side, which is the hypotenuse. The legs of an isosceles right triangle are equal in length, and the hypotenuse is equal to the square root of the sum of the squares of the legs. This can be represented by the equation $c^2 = a^2 + b^2$ , where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the legs.

Analyzing the Statements

A. $\overline{XZ}+\overline{XY}$

This statement is not sufficient to prove that $\Delta XYZ$ is an isosceles right triangle. The sum of the lengths of two sides of a triangle does not necessarily mean that the triangle is an isosceles right triangle. For example, a triangle with sides of length 3, 4, and 5 is a right triangle, but it is not an isosceles right triangle.

B. $XZ = XY = 5$

This statement is sufficient to prove that $\Delta XYZ$ is an isosceles right triangle. Since the two legs of the triangle are equal in length, and the hypotenuse is also equal to the square root of the sum of the squares of the legs, we can conclude that $\Delta XYZ$ is an isosceles right triangle.

C. The slope of $\overline{XZ}$ is $\frac{3}{4}$ , the slope of $\overline{XY}$

This statement is not sufficient to prove that $\Delta XYZ$ is an isosceles right triangle. The slope of a line is a measure of how steep it is, and it does not necessarily relate to the length of the sides of a triangle. For example, a line with a slope of $\frac{3}{4}$ can have any length, and it does not necessarily mean that the triangle is an isosceles right triangle.

In conclusion, the statement that proves that $\Delta XYZ$ is an isosceles right triangle is $XZ = XY = 5$ . This statement is sufficient to prove that the triangle is an isosceles right triangle, since the two legs of the triangle are equal in length, and the hypotenuse is also equal to the square root of the sum of the squares of the legs.

Suppose we have a triangle $\Delta XYZ$ with sides of length 3, 4, and 5. We can use the Pythagorean theorem to check if the triangle is a right triangle. The Pythagorean theorem states that $c^2 = a^2 + b^2$ , where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the legs. In this case, we have $c^2 = 5^2 = 25$ , and $a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$ . Since $c^2 = a^2 + b^2$ , we can conclude that the triangle is a right triangle.

However, we cannot conclude that the triangle is an isosceles right triangle, since the two legs of the triangle are not equal in length. Therefore, the statement $XZ = XY = 5$ is the only statement that proves that $\Delta XYZ$ is an isosceles right triangle.

An isosceles right triangle has two equal sides, which are the legs of the triangle, and the third side, which is the hypotenuse.
The statement $XZ = XY = 5$ is sufficient to prove that $\Delta XYZ$ is an isosceles right triangle.
The Pythagorean theorem can be used to check if a triangle is a right triangle.
The slope of a line is a measure of how steep it is, and it does not necessarily relate to the length of the sides of a triangle.
Frequently Asked Questions (FAQs) About Isosceles Right Triangles ====================================================================

Q: What is an isosceles right triangle?

A: An isosceles right triangle is a special type of triangle where two sides are equal in length, and the third side is the hypotenuse, which is equal to the square root of the sum of the squares of the other two sides.

Q: What are the characteristics of an isosceles right triangle?

A: The characteristics of an isosceles right triangle are:

Two equal sides, which are the legs of the triangle
The third side, which is the hypotenuse, is equal to the square root of the sum of the squares of the other two sides
The legs of the triangle are perpendicular to each other

Q: How do I know if a triangle is an isosceles right triangle?

A: To determine if a triangle is an isosceles right triangle, you can use the following methods:

Check if the two legs of the triangle are equal in length
Check if the hypotenuse is equal to the square root of the sum of the squares of the other two sides
Use the Pythagorean theorem to check if the triangle is a right triangle

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides. The formula is:

c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Q: How do I use the Pythagorean theorem to check if a triangle is a right triangle?

A: To use the Pythagorean theorem to check if a triangle is a right triangle, you can follow these steps:

Identify the lengths of the three sides of the triangle
Plug the lengths of the sides into the Pythagorean theorem formula
Check if the equation is true
If the equation is true, then the triangle is a right triangle

Q: What is the difference between an isosceles right triangle and a right triangle?

A: The difference between an isosceles right triangle and a right triangle is that an isosceles right triangle has two equal sides, which are the legs of the triangle, while a right triangle does not have equal sides.

Q: Can a right triangle be an isosceles right triangle?

A: Yes, a right triangle can be an isosceles right triangle if it has two equal sides, which are the legs of the triangle.

Q: Can an isosceles right triangle be a scalene triangle?

A: No, an isosceles right triangle cannot be a scalene triangle, because a scalene triangle has three unequal sides, while an isosceles right triangle has two equal sides.

Q: Can an isosceles right triangle be an equilateral triangle?

A: No, an isosceles right triangle cannot be an equilateral triangle, because an equilateral triangle has three equal sides, while an isosceles right triangle has two equal sides and one unequal side.

Q: Can an isosceles right triangle be a triangle with a 90-degree angle?

A: Yes, an isosceles right triangle can be a triangle with a 90-degree angle, because the two legs of the triangle are perpendicular to each other.

Q: Can an isosceles right triangle be a triangle with a 45-degree angle?

A: No, an isosceles right triangle cannot be a triangle with a 45-degree angle, because the two legs of the triangle are perpendicular to each other, and a 45-degree angle is not perpendicular to the other angle.