Keysha is building a coffee table. It’s going to be in the shape of a rectangular prism, without the faces. She wants to make the table decorative by adding metal diagonals. The length of the coffee table will be 44 in., the width 24 in., and the height 14 in. Find the length Keysha should make each diagonal. Round the answer to the nearest tenth.(1 point)
in.
To find the length of each diagonal, we can use the Pythagorean theorem.
Let's consider the diagonal on the bottom face of the coffee table, which forms a right triangle with the length and width of the table.
Using the Pythagorean theorem, we have:
diagonal^2 = length^2 + width^2
diagonal^2 = 44^2 + 24^2
diagonal^2 = 1936 + 576
diagonal^2 = 2512
diagonal ≈ √2512 ≈ 50.12 in
So, Keysha should make each diagonal approximately 50.1 inches long.
Use the image to answer the question.
A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image.
(4 points)
In the given image, we have a square made up of four right-angled triangles. We can see that each right-angled triangle has two sides labeled "a" and "b" and a hypotenuse labeled "c".
The sides "a" and "b" are positioned in such a way that they form the four outer corners of the square, with the right angles at those corners.
Now, if we focus on the inner square formed by the intersection of the four right-angled triangles, we can see that each side of the inner square is labeled "c".
Now, let's consider one of the four right-angled triangles. The sides "a" and "b" of this triangle are the shorter and longer parts of one side of the inner square, respectively. And the hypotenuse "c" of this triangle is actually the diagonal of the inner square.
According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Therefore, we can apply the Pythagorean theorem to each of the four right-angled triangles in the image. Specifically, we have:
c^2 = a^2 + b^2
Now, as we can see, the four triangles have the same dimensions, so the lengths of the sides "a" and "b" in each triangle are the same. Therefore, we can rewrite the equation as:
c^2 = a^2 + a^2
c^2 = 2a^2
Taking the square root of both sides gives us:
c = √(2a^2)
c = a√2
So, we can conclude that the length of each side of the inner square, labeled "c", is equal to the length of the sides of the outer square, labeled "a", multiplied by the square root of 2.
This result is consistent with the Pythagorean theorem and provides a proof of the theorem using the given image.