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
46.6 cm
46.6 cm
59.5 cm
59.5 cm
51.2 cm
51.2 cm
54.4 cm
To find the length of the diagonal BE of the rectangular prism, we can use the Pythagorean theorem.
First, we need to find the length of side HE. Using the Pythagorean theorem in the right triangle HEB:
HE^2 + BH^2 = BE^2
HE^2 + 24^2 = 40^2
HE^2 + 576 = 1600
HE^2 = 1024
HE = √1024
HE = 32
Now, we can find the length of the diagonal BE:
BE^2 = BH^2 + HE^2
BE^2 = 40^2 + 32^2
BE^2 = 1600 + 1024
BE^2 = 2624
BE = √2624
BE ≈ 51.2 cm
Therefore, the length of the diagonal of the rectangular prism, segment BE, is approximately 51.2 cm.
Correct response:
51.2 cm