In this lesson, we will extend the concept of the Pythagorean theorem into three dimensions. 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 three dimensions, we will be working with right prisms and right pyramids. A right prism is a solid figure with two congruent, parallel bases and rectangular sides. A right pyramid is a solid figure with a polygonal base and triangular sides that meet at a common vertex.
To apply the Pythagorean theorem in three dimensions, we need to consider the diagonals or height of these solid figures. In a right prism, the height is the perpendicular distance between the bases. In a right pyramid, the height is the perpendicular distance from the base to the vertex.
To find the length of the height or diagonal, we can use the Pythagorean theorem. Let's take a right prism as an example. If we have a right prism with a base of length a, a width of length b, and a height of length c, the diagonal (d) can be found using the formula:
d = √(a^2 + b^2 + c^2)
Similarly, in a right pyramid, if we have a base of length a, a height of length b, and the diagonal from the base to the top vertex is c, we can find the height using the formula:
b = √(c^2 - a^2)
It's important to note that these formulas only apply to right prisms and right pyramids. If the solid figure is not a right prism or right pyramid, the Pythagorean theorem may not be applicable.
Now you have learned how to apply the Pythagorean theorem in three dimensions!
Math 8 A
Unit 6
Triangles
Lesson 7
The Pythagorean theorem in three dimensions
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