A carpenter is building rectangular walls for a room addition. The width of a section of wall is two times the height h. Each section has a brace that connects two opposite corners of the section. What is a simplified expression for the length of a brace?
Is this written correctly? Usually the height is longer than the width! Again, draw a picture. Another right triangle.
Let x = height; let 2x = width. Using Pythagorean Theorem:
hyp^2 = x^2 + (2x)^2
hyp^2 = x^2 + 4x^2
hyp^2 = 5x^2
take square root of each side:
hyp = x * sqrt(5)
Why did the brace go to therapy? Because it had an acute angle problem!
Now, let's tackle your question. We can use the Pythagorean theorem to find the length of the brace. The theorem states that the square of the hypotenuse (in this case, the brace) is equal to the sum of the squares of the other two sides (the height h and the width 2h).
So, if the brace is represented by the letter b, we can write the equation:
b² = h² + (2h)²
Simplifying this equation, we get:
b² = h² + 4h²
b² = 5h²
Taking the square root of both sides, we find:
b = √(5h²)
So, the simplified expression for the length of the brace is √(5h²). However, if the carpenter needs a numerical value, we can simplify further by observing that √(5h²) is equal to √5 * √h². Therefore, the simplified expression would be √5h.
To find the length of the brace, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the sections of wall form a right triangle, with the height (h) as one side and the width (2h) as the other side. The brace connects the two opposite corners of the section, which is the hypotenuse of the right triangle.
Using the Pythagorean theorem, we can write the equation as:
Length of the brace^2 = height^2 + width^2
Substituting the values, we get:
Length of the brace^2 = h^2 + (2h)^2 = h^2 + 4h^2 = 5h^2
Taking the square root of both sides to solve for the length of the brace, we have:
Length of the brace = √(5h^2)
Simplifying, we get:
Length of the brace = √5 * h
Therefore, the simplified expression for the length of a brace is √5 * h.
To find the length of a brace, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side, which is called the hypotenuse.
In this case, we can consider the brace as the hypotenuse of a right-angled triangle, with the height (h) as one of the shorter sides. The other shorter side is the width, which is twice the height. So, the width would be 2h.
Let's label the length of the brace as "b." According to the Pythagorean theorem, the equation would be:
b^2 = h^2 + (2h)^2
Simplifying this equation, we have:
b^2 = h^2 + 4h^2
b^2 = 5h^2
To find a simplified expression for the length of the brace, we can take the square root of both sides of the equation:
√(b^2) = √(5h^2)
b = √5h
Therefore, the simplified expression for the length of the brace is √5h.