Consider The Incomplete Paragraph Proof.Given: Isosceles Right Triangle XYZ { (45 {\circ}-45 \circ}-90^{\circ}$}$ Triangle)Prove In A ${$45^{\circ -45 {\circ}-90 {\circ}$}$ Triangle, The Hypotenuse Is { \sqrt{2}$}$ Times

The Incomplete Paragraph Proof: A Step-by-Step Guide to Proving the Hypotenuse of an Isosceles Right Triangle

In mathematics, a right triangle is a fundamental concept that has been studied for centuries. One of the most interesting properties of a right triangle is the relationship between its sides. In this article, we will explore the property of an isosceles right triangle, specifically the relationship between the hypotenuse and the other two sides. We will use the given information to prove that the hypotenuse of an isosceles right triangle is $\sqrt{2}$ times the length of one of the legs.

We are given an isosceles right triangle XYZ with angles $45^{\circ}-45^{\circ}-90^{\circ}$. This means that the two legs of the triangle, XY and XZ, are equal in length, and the hypotenuse, YZ, is opposite the right angle.

We need to prove that the hypotenuse, YZ, is $\sqrt{2}$ times the length of one of the legs, say XY. In other words, we need to show that $YZ = \sqrt{2} \cdot XY$.

Step 1: Draw the Altitude

To start the proof, we will draw the altitude from the right angle, X, to the hypotenuse, YZ. Let's call the point where the altitude intersects the hypotenuse, H.

Step 2: Identify Similar Triangles

By drawing the altitude, we have created two similar triangles, XHY and XHZ. These triangles are similar because they have the same angles.

Step 3: Use the Properties of Similar Triangles

Since the triangles are similar, we can use the properties of similar triangles to relate the sides of the triangles. Specifically, we can use the fact that the corresponding sides of similar triangles are proportional.

Step 4: Set Up the Proportion

Let's set up a proportion using the corresponding sides of the similar triangles. We can write:

$$\frac{XY}{XZ} = \frac{XH}{HZ}$$

Step 5: Simplify the Proportion

Since the triangles are isosceles, we know that $XY = XZ$. Therefore, we can simplify the proportion to:

$$\frac{XH}{HZ} = 1$$

Step 6: Use the Pythagorean Theorem

Now, we can use the Pythagorean theorem to relate the sides of the triangle. Specifically, we can write:

$$HZ^2 + XH^2 = XY^2$$

Step 7: Substitute the Proportion

We can substitute the proportion we derived earlier into the Pythagorean theorem:

$$HZ^2 + (HZ)^2 = XY^2$$

Step 8: Simplify the Equation

Simplifying the equation, we get:

$$2HZ^2 = XY^2$$

Step 9: Take the Square Root

Taking the square root of both sides, we get:

$$HZ = \frac{XY}{\sqrt{2}}$$

Step 10: Conclude the Proof

Therefore, we have shown that the hypotenuse, YZ, is $\sqrt{2}$ times the length of one of the legs, say XY. This completes the proof.

In this article, we have used the given information to prove that the hypotenuse of an isosceles right triangle is $\sqrt{2}$ times the length of one of the legs. We have used the properties of similar triangles and the Pythagorean theorem to derive the relationship between the sides of the triangle. This proof is a classic example of how mathematical concepts can be used to derive interesting and useful results.

For more information on the properties of right triangles, including the Pythagorean theorem, see the following resources:

• Wikipedia: Pythagorean Theorem
• Math Open Reference: Pythagorean Theorem
• Khan Academy: Pythagorean Theorem

The proof of the hypotenuse of an isosceles right triangle is a classic example of how mathematical concepts can be used to derive interesting and useful results. By using the properties of similar triangles and the Pythagorean theorem, we have shown that the hypotenuse is $\sqrt{2}$ times the length of one of the legs. This proof is a great example of how mathematical concepts can be used to solve real-world problems.
Frequently Asked Questions: The Hypotenuse of an Isosceles Right Triangle

Q: What is an isosceles right triangle?

A: An isosceles right triangle is a type of right triangle where the two legs are equal in length, and the hypotenuse is opposite the right angle.

Q: What is the relationship between the hypotenuse and the legs of an isosceles right triangle?

A: The hypotenuse of an isosceles right triangle is $\sqrt{2}$ times the length of one of the legs.

Q: How do you prove that the hypotenuse is $\sqrt{2}$ times the length of one of the legs?

A: To prove that the hypotenuse is $\sqrt{2}$ times the length of one of the legs, you can use the properties of similar triangles and the Pythagorean theorem. Specifically, you can draw the altitude from the right angle to the hypotenuse, identify similar triangles, and use the properties of similar triangles to relate the sides of the triangles.

Q: What is the Pythagorean theorem?

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

Q: How do you use the Pythagorean theorem to relate the sides of a right triangle?

A: To use the Pythagorean theorem to relate the sides of a right triangle, you can write an equation using the formula $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the other two sides, and $c$ is the length of the hypotenuse.

Q: What is the significance of the $\sqrt{2}$ factor in the relationship between the hypotenuse and the legs of an isosceles right triangle?

A: The $\sqrt{2}$ factor is significant because it represents the ratio of the length of the hypotenuse to the length of one of the legs. This ratio is a fundamental property of isosceles right triangles and is used in a wide range of mathematical and scientific applications.

Q: How do you apply the relationship between the hypotenuse and the legs of an isosceles right triangle in real-world problems?

A: The relationship between the hypotenuse and the legs of an isosceles right triangle can be applied in a wide range of real-world problems, including:

• Calculating the length of a hypotenuse in a right triangle
• Finding the length of one of the legs in a right triangle
• Solving problems involving similar triangles
• Calculating distances and heights in engineering and architecture

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:

• Failing to recognize that the two legs are equal in length
• Failing to use the properties of similar triangles to relate the sides of the triangles
• Failing to apply the Pythagorean theorem correctly
• Failing to check for errors in calculations

Q: How do you check for errors in calculations involving isosceles right triangles?

A: To check for errors in calculations involving isosceles right triangles, you can:

• Double-check your calculations for accuracy
• Use a calculator or computer program to verify your results
• Check your work against a known solution or reference
• Ask a colleague or teacher to review your work

In this article, we have answered some of the most frequently asked questions about the hypotenuse of an isosceles right triangle. We have discussed the relationship between the hypotenuse and the legs of an isosceles right triangle, how to prove that the hypotenuse is $\sqrt{2}$ times the length of one of the legs, and how to apply this relationship in real-world problems. We have also discussed common mistakes to avoid and how to check for errors in calculations.