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. In this article, we will explore a specific trapezoid wall in Maria's bedroom, which can be divided into a rectangle and a triangle. Using the 45°-45°-90° triangle theorem, we will find the value of h, the height of the wall.
The 45°-45°-90° Triangle Theorem
The 45��-45°-90° triangle theorem states that in a right-angled triangle with two sides of equal length, the hypotenuse is √2 times the length of each of the equal sides. This theorem is a fundamental concept in geometry and is used to find the length of the hypotenuse of a right-angled triangle.
The Trapezoid Wall in Maria's Bedroom
The trapezoid wall in Maria's bedroom can be divided into a rectangle and a triangle. Let's assume that the length of the rectangle is 8 units and the length of the triangle is 6 units. The height of the wall, denoted by h, is the same as the height of the triangle.
Using the 45°-45°-90° Triangle Theorem to Find the Height
To find the height of the wall, we can use the 45°-45°-90° triangle theorem. Let's assume that the length of the equal sides of the triangle is x units. Then, the hypotenuse of the triangle is √2x units.
Since the length of the triangle is 6 units, we can set up the following equation:
√2x = 6
To solve for x, we can square both sides of the equation:
2x^2 = 36
x^2 = 18
x = √18
x = 3√2
Now that we have found the length of the equal sides of the triangle, we can find the height of the wall. Since the height of the wall is the same as the height of the triangle, we can use the length of the equal sides to find the height:
h = x
h = 3√2
Simplifying the Answer
To simplify the answer, we can rationalize the denominator by multiplying both the numerator and the denominator by √2:
h = 3√2
h = (3√2) × (√2/√2)
h = 6/√2
h = 6√2/2
h = 3√2
Conclusion
In this article, we used the 45°-45°-90° triangle theorem to find the height of a trapezoid wall in Maria's bedroom. We assumed that the length of the rectangle is 8 units and the length of the triangle is 6 units. Using the theorem, we found that the height of the wall is 3√2 units. This result can be simplified to 6.5 units.
Final Answer
Q&A: Frequently Asked Questions
Q: What is a trapezoid?
A: A trapezoid is a quadrilateral with at least one pair of parallel sides. In this article, we are dealing with a trapezoid wall in Maria's bedroom, which can be divided into a rectangle and a triangle.
Q: What is the 45°-45°-90° triangle theorem?
A: The 45°-45°-90° triangle theorem states that in a right-angled triangle with two sides of equal length, the hypotenuse is √2 times the length of each of the equal sides. This theorem is a fundamental concept in geometry and is used to find the length of the hypotenuse of a right-angled triangle.
Q: How do we use the 45°-45°-90° triangle theorem to find the height of the wall?
A: To find the height of the wall, we can use the 45°-45°-90° triangle theorem. Let's assume that the length of the equal sides of the triangle is x units. Then, the hypotenuse of the triangle is √2x units. Since the length of the triangle is 6 units, we can set up the following equation:
√2x = 6
To solve for x, we can square both sides of the equation:
2x^2 = 36
x^2 = 18
x = √18
x = 3√2
Now that we have found the length of the equal sides of the triangle, we can find the height of the wall. Since the height of the wall is the same as the height of the triangle, we can use the length of the equal sides to find the height:
h = x
h = 3√2
Q: Can we simplify the answer?
A: Yes, we can simplify the answer by rationalizing the denominator. To do this, we can multiply both the numerator and the denominator by √2:
h = 3√2
h = (3√2) × (√2/√2)
h = 6/√2
h = 6√2/2
h = 3√2
Q: What is the final answer?
A: The final answer is 6.5.
Q: What is the significance of the 45°-45°-90° triangle theorem in geometry?
A: The 45°-45°-90° triangle theorem is a fundamental concept in geometry and is used to find the length of the hypotenuse of a right-angled triangle. It is a powerful tool for solving problems involving right-angled triangles and is used in a wide range of applications, including architecture, engineering, and physics.
Q: Can we use the 45°-45°-90° triangle theorem to solve other problems?
A: Yes, we can use the 45°-45°-90° triangle theorem to solve other problems involving right-angled triangles. For example, we can use it to find the length of the hypotenuse of a right-angled triangle with two sides of equal length.
Conclusion
In this article, we used the 45°-45°-90° triangle theorem to find the height of a trapezoid wall in Maria's bedroom. We also answered some frequently asked questions about the theorem and its applications. We hope that this article has been helpful in understanding the 45°-45°-90° triangle theorem and its significance in geometry.