The Pythagorean Theorem in Three Dimensions Quick Check
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Question
Use the image to answer the question.
An illustration of a rectangular prism is shown with labeled parts. The left and back base edges are not visible but their positions are indicated by a dashed line. The 6 vertices are labeled A B C D E F and G. The base edge A B is labeled length. Base edge B C is labeled width. The perpendicular C D is labeled height. A dotted line crosses the rectangular base through the center to show diagonal B H. A dashed line extends up from the base indicating the left rear perpendicular E H. A dotted line crosses from the back top left vertex E to the front bottom right vertex B to show diagonal B E. This dotted line is labeled diagonal.
The length of the prism is 32 cm, the width is 24 cm, the height is 44 cm, and the length of the diagonal of the base, segment BH , is 40 cm. Find the length of the diagonal of the rectangular prism, segment BE . Round the answer to the nearest tenth.
(1 point)
Responses
59.5 cm
59.5 cm
46.6 cm
46.6 cm
51.2 cm
51.2 cm
54.4 cm
54.4 cm
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To find the length of diagonal BE, we can use the Pythagorean theorem. Diagonal BE is the hypotenuse of a right triangle formed by segment BH, which is one leg, and a segment perpendicular to BH, which is the other leg.
Using the Pythagorean theorem, we can write the equation as:
BE^2 = BH^2 + EH^2
Since BH is given as 40 cm and EH is the height of the prism which is 44 cm, we can substitute these values into the equation:
BE^2 = 40^2 + 44^2
Simplifying, we get:
BE^2 = 1600 + 1936
BE^2 = 3536
Taking the square root of both sides to solve for BE, we get:
BE ≈ √3536
BE ≈ 59.5 cm
Therefore, the length of the diagonal of the rectangular prism, segment BE, is approximately 59.5 cm.