Question

The Pythagorean Theorem in Three Dimensions Practice

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An illustration shows a cone that rests flat on the circular base. The visible circumference of the base and sides are in solid lines. The circumference of the base at the back that is not visible is indicated by a dashed line. Parts are labeled. A large point is marked in the center of the base. A solid line passes through this point and extends from one edge of the base to the other to indicate the diameter of the base. The line is labeled d equals 10 centimeters. The right side of the cone extending from where the diameter meets the edge of the base up to the point of the cone shows a slant height of 13 centimeters. A dotted line extends from the interior of the vertex of the cone down to the point marked in the center of the base. A right angle is indicated where the dotted height line meets the diameter line in the center of the base.

Find the length of the height of the cone.

(1 point)
cm

Answer 1

[AI-generated answer — Bot GPT 3.5]

Based on the given information, we can use the Pythagorean Theorem to find the height of the cone. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the height (h) of the cone is one of the sides, and the slant height (13 cm) and the radius (half of the diameter, which is 10/2 = 5 cm) are the other two sides.

Using the Pythagorean Theorem, we have:

h^2 = (slant height)^2 - (radius)^2
h^2 = 13^2 - 5^2
h^2 = 169 - 25
h^2 = 144

Taking the square root of both sides, we get:

h = √144
h = 12 cm.

Therefore, the length of the height of the cone is 12 cm.