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)
To find the length of the diagonal of the prism, we can use the Pythagorean theorem.
First, we find the length of the diagonal of the front face. This can be found by using the width and height of the front face as the two sides of a right triangle.
Using the Pythagorean theorem, we have:
diagonal of front face squared = width squared + height squared
diagonal of front face squared = 3^2 + 12^2
diagonal of front face squared = 9 + 144
diagonal of front face squared = 153
Taking the square root of both sides, we find:
diagonal of front face = sqrt(153)
Now, we find the length of the diagonal of the right face by using the length and height of the right face as the two sides of a right triangle.
Using the Pythagorean theorem, we have:
diagonal of right face squared = length squared + height squared
diagonal of right face squared = 4^2 + 12^2
diagonal of right face squared = 16 + 144
diagonal of right face squared = 160
Taking the square root of both sides, we find:
diagonal of right face = sqrt(160)
Finally, we can find the length of the diagonal of the prism by using the diagonal of the front face, the diagonal of the right face, and the given diagonal inside the prism.
Using the Pythagorean theorem, we have:
length of diagonal of prism squared = diagonal of front face squared + diagonal of right face squared + interior diagonal squared
length of diagonal of prism squared = (sqrt(153))^2 + (sqrt(160))^2 + interior diagonal squared
length of diagonal of prism squared = 153 + 160 + interior diagonal squared
We are not given the length of the interior diagonal, so it is not possible to find the exact length of the diagonal of the prism with the information given in the image.