Triangles Unit Test
10 of 1510 of 15 Items
Question
Use the image to answer the question.
A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 9. Data points marked are at left parenthesis 3 comma 2 right parenthesis, and left parenthesis 7 comma 8 right parenthesis. A straight line joins the two points forming a positive slope.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth.
(1 point)
$$
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To find the length between the two points, we use the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
In this case, (x1, y1) = (3, 2) and (x2, y2) = (7, 8).
Substituting these values into the formula, we get:
Distance = sqrt((7 - 3)^2 + (8 - 2)^2)
= sqrt(4^2 + 6^2)
= sqrt(16 + 36)
= sqrt(52)
≈ 7.21 (rounded to the nearest hundredth)
Therefore, the length between the two points is approximately 7.21.
Use the image to answer the question.
An illustration shows a square pyramid with labeled parts. The front and left faces of the pyramid and base are visible. The back and right faces are not visible. The back and right base edges are not visible but their position is indicated by a dashed line. The left base edge shows a length of 16 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and shows a height of 15. A right angle is indicated where the dotted height line meets the base.
Find the slant height of the square pyramid.
(1 point)
units
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To find the slant height of the square pyramid, we can use the Pythagorean Theorem.
In this case, we have a right triangle formed by the height, the slant height, and the length of the base edge. The height (h) and the base edge length (b) are given as 15 feet and 16 feet, respectively.
Using the Pythagorean Theorem, we have:
slant height^2 = height^2 + base edge^2
slant height^2 = 15^2 + 16^2
slant height^2 = 225 + 256
slant height^2 = 481
Taking the square root of both sides, we get:
slant height = √481
≈ 21.93 feet (rounded to the nearest hundredth)
Therefore, the slant height of the square pyramid is approximately 21.93 feet.
Use the image to answer the question.
An illustration of a rectangular prism is shown with labeled parts. The top, front and right sides are visible. The bottom, left, and rear sides are not visible, but their positions are indicated by dashed lines. The vertical edge joining the front and left sides shows a height of 12 centimeters. The front base edge shows a width of 3 centimeters. The right base edge shows a length of 4 centimeters. A dotted line extends diagonally from the vertex at the bottom back right side of the prism through the interior of the prism to the vertex at the top front left.
Find the length of the diagonal of the prism.
(1 point)
cm
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To find the length of the diagonal of the rectangular prism, we can use the Pythagorean Theorem.
In this case, we have a right triangle formed by the height, the width, and the length of the prism. The height (h) is given as 12 centimeters, the width (w) is given as 3 centimeters, and the length (l) is given as 4 centimeters.
Using the Pythagorean Theorem, we have:
diagonal^2 = height^2 + width^2 + length^2
diagonal^2 = 12^2 + 3^2 + 4^2
diagonal^2 = 144 + 9 + 16
diagonal^2 = 169
Taking the square root of both sides, we get:
diagonal = √169
= 13 centimeters
Therefore, the length of the diagonal of the prism is 13 centimeters.
The Great Pyramid of Giza in Egypt has a square base with edge length of 230.5 m and a slant height of 180.2 m. Find the height of the pyramid. Round the answer to the nearest tenth.(1 point)
m
To find the height of the pyramid, we can use the Pythagorean Theorem.
In this case, we have a right triangle formed by the height, half the base, and the slant height of the pyramid. The base length is given as 230.5 m and the slant height is given as 180.2 m.
Using the Pythagorean Theorem, we have:
height^2 = slant height^2 - (base length/2)^2
height^2 = 180.2^2 - (230.5/2)^2
height^2 = 32546.04 - 26554.06
height^2 = 5991.98
Taking the square root of both sides, we get:
height = √5991.98
height ≈ 77.4 m (rounded to the nearest tenth)
Therefore, the height of the pyramid is approximately 77.4 m.
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.
In this case, we have a right triangle formed by the length, width, and height of the coffee table. The length is given as 44 inches, the width is given as 24 inches, and the height is given as 14 inches.
Using the Pythagorean Theorem, we have:
diagonal^2 = length^2 + width^2 + height^2
diagonal^2 = 44^2 + 24^2 + 14^2
diagonal^2 = 1936 + 576 + 196
diagonal^2 = 2708
Taking the square root of both sides, we get:
diagonal = √2708
diagonal ≈ 52 inches (rounded to the nearest tenth)
Therefore, Keysha should make each diagonal approximately 52 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.