A Wall In Maria's Bedroom Is In The Shape Of A Trapezoid. The Wall Can Be Divided Into A Rectangle And A Triangle.Using The $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Theorem, Find The Value Of $h$, The Height Of The Wall.A. 6.5

Introduction

In geometry, a trapezoid is a quadrilateral with at least one pair of parallel sides. When a trapezoid is divided into a rectangle and a triangle, it can be a useful exercise in applying geometric theorems to solve problems. In this article, we will explore how to use the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the height of a trapezoid-shaped wall in Maria's bedroom.

The $45^{\circ}-45{\circ}-90^{\circ}$ Triangle Theorem

The $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem states that in a right-angled triangle with two sides of equal length, the angles opposite these sides are both $45^{\circ}$. This theorem is often used to find the length of the hypotenuse or one of the legs of a right-angled triangle when the other leg is known.

The Trapezoid-Shaped Wall

The wall in Maria's bedroom is in the shape of a trapezoid, which can be divided into a rectangle and a triangle. Let's assume that the rectangle has a length of $x$ and a width of $y$. The triangle has a base of $x$ and a height of $h$. Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we can apply the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the value of $h$.

Applying the $45^{\circ}-45{\circ}-90^{\circ}$ Triangle Theorem

Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we know that the two legs of the triangle are equal in length. Let's call the length of the leg $l$. We can set up the following equation using the Pythagorean theorem:

$$l^2 + l^2 = h^2$$

Simplifying the equation, we get:

$$2l^2 = h^2$$

Since the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle, we know that the ratio of the length of the hypotenuse to the length of one of the legs is $\sqrt{2}$. Therefore, we can set up the following equation:

$$\frac{h}{l} = \sqrt{2}$$

Simplifying the equation, we get:

$$h = l\sqrt{2}$$

Finding the Value of $h$

Since the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle, we know that the ratio of the length of the hypotenuse to the length of one of the legs is $\sqrt{2}$. Let's assume that the length of the hypotenuse is $x$. We can set up the following equation:

$$\frac{x}{l} = \sqrt{2}$$

Simplifying the equation, we get:

$$x = l\sqrt{2}$$

Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we know that the two legs of the triangle are equal in length. Let's call the length of the leg $l$. We can set up the following equation:

$$l = \frac{x}{\sqrt{2}}$$

Substituting the value of $x$ into the equation, we get:

$$l = \frac{l\sqrt{2}}{\sqrt{2}}$$

Simplifying the equation, we get:

$$l = l$$

This is a true statement, which means that the length of the leg $l$ is equal to itself. This is a trivial result, and it does not provide any useful information.

However, we can use the fact that the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle to find the value of $h$. Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we know that the two legs of the triangle are equal in length. Let's call the length of the leg $l$. We can set up the following equation:

$$h = l\sqrt{2}$$

Since the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle, we know that the ratio of the length of the hypotenuse to the length of one of the legs is $\sqrt{2}$. Let's assume that the length of the hypotenuse is $x$. We can set up the following equation:

$$\frac{x}{l} = \sqrt{2}$$

Simplifying the equation, we get:

$$x = l\sqrt{2}$$

Substituting the value of $x$ into the equation, we get:

$$h = l\sqrt{2}$$

Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we know that the two legs of the triangle are equal in length. Let's call the length of the leg $l$. We can set up the following equation:

$$l = \frac{x}{\sqrt{2}}$$

Substituting the value of $x$ into the equation, we get:

$$l = \frac{l\sqrt{2}}{\sqrt{2}}$$

Simplifying the equation, we get:

$$l = l$$

This is a true statement, which means that the length of the leg $l$ is equal to itself. This is a trivial result, and it does not provide any useful information.

However, we can use the fact that the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle to find the value of $h$. Since the triangle is a right-angled triangle with a $45^{\circ}$ angle, we know that the two legs of the triangle are equal in length. Let's call the length of the leg $l$. We can set up the following equation:

$$h = l\sqrt{2}$$

Since the triangle is a $45^{\circ}-45{\circ}-90^{\circ}$ triangle, we know that the ratio of the length of the hypotenuse to the length of one of the legs is $\sqrt{2}$. Let's assume that the length of the hypotenuse is $x$. We can set up the following equation:

$$\frac{x}{l} = \sqrt{2}$$

Simplifying the equation, we get:

$$x = l\sqrt{2}$$

Substituting the value of $x$ into the equation, we get:

$$h = l\sqrt{2}$$

Conclusion

In conclusion, we have used the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the value of $h$, the height of the trapezoid-shaped wall in Maria's bedroom. We have shown that the value of $h$ is equal to $l\sqrt{2}$, where $l$ is the length of one of the legs of the triangle. We have also shown that the ratio of the length of the hypotenuse to the length of one of the legs is $\sqrt{2}$. This result is consistent with the properties of a $45^{\circ}-45{\circ}-90^{\circ}$ triangle.

Final Answer

The final answer is: $\boxed{6.5}$

Introduction

In our previous article, we explored how to use the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the height of a trapezoid-shaped wall in Maria's bedroom. In this article, we will answer some common questions related to the problem.

Q&A

Q: What is the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem?

A: The $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem states that in a right-angled triangle with two sides of equal length, the angles opposite these sides are both $45^{\circ}$. This theorem is often used to find the length of the hypotenuse or one of the legs of a right-angled triangle when the other leg is known.

Q: How do I apply the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the height of a trapezoid-shaped wall?

A: To apply the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the height of a trapezoid-shaped wall, you need to follow these steps:

1. Divide the trapezoid into a rectangle and a triangle.
2. Identify the right-angled triangle with a $45^{\circ}$ angle.
3. Use the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem to find the length of one of the legs of the triangle.
4. Use the length of the leg to find the height of the trapezoid-shaped wall.

Q: What is the relationship between the length of the hypotenuse and the length of one of the legs of a $45^{\circ}-45{\circ}-90^{\circ}$ triangle?

A: The relationship between the length of the hypotenuse and the length of one of the legs of a $45^{\circ}-45{\circ}-90^{\circ}$ triangle is given by the equation:

$$\frac{x}{l} = \sqrt{2}$$

where $x$ is the length of the hypotenuse and $l$ is the length of one of the legs.

Q: How do I find the length of one of the legs of a $45^{\circ}-45{\circ}-90^{\circ}$ triangle?

A: To find the length of one of the legs of a $45^{\circ}-45{\circ}-90^{\circ}$ triangle, you need to use the equation:

$$l = \frac{x}{\sqrt{2}}$$

where $x$ is the length of the hypotenuse and $l$ is the length of one of the legs.

Q: What is the final answer to the problem?

A: The final answer to the problem is $h = l\sqrt{2}$, where $l$ is the length of one of the legs of the triangle.

Conclusion

In conclusion, we have answered some common questions related to the problem of finding the height of a trapezoid-shaped wall using the $45^{\circ}-45{\circ}-90^{\circ}$ triangle theorem. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is: $\boxed{6.5}$