Finding All Three Sides of a Triangle with Vertices P(2, -1, 0), Q(4, 1, 1), and R(4, -5, 4)
In this article, we will learn how to find all three sides of a triangle given the coordinates of its three vertices in 3D space. This is a common problem in high school geometry and can be solved using the distance formula.
To begin, let's define the three vertices of the triangle:
- Vertex P: (2, -1, 0)
- Vertex Q: (4, 1, 1)
- Vertex R: (4, -5, 4)
Step 1: Find the length of side PQ
To find the length of side PQ, we need to use the distance formula:
Distance between two points (x1, y1, z1) and (x2, y2, z2) = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]
Plugging in the coordinates of P and Q, we get:
Distance PQ = √[(4 - 2)^2 + (1 - (-1))^2 + (1 - 0)^2] = √[4^2 + 2^2 + 1^2] = √(16 + 4 + 1) = √21
Therefore, the length of side PQ is √21.
Step 2: Find the length of side PR
To find the length of side PR, we need to use the distance formula again:
Distance between P(2, -1, 0) and R(4, -5, 4) = √[(4 - 2)^2 + (-5 - (-1))^2 + (4 - 0)^2] = √[4^2 + (-4)^2 + 4^2] = √(16 + 16 + 16) = √48 = 2√12
Therefore, the length of side PR is 2√12.
Step 3: Find the length of side QR
To find the length of side QR, we need to use the distance formula one more time:
Distance between Q(4, 1, 1) and R(4, -5, 4) = √[(4 - 4)^2 + (-5 - 1)^2 + (4 - 1)^2] = √[0^2 + (-6)^2 + 3^2] = √(36 + 9) = √45
Therefore, the length of side QR is √45.
In conclusion, the three sides of the triangle with vertices P(2, -1, 0), Q(4, 1, 1), and R(4, -5, 4) are:
- Side PQ: √21
- Side PR: 2√12
- Side QR: √45
By using the distance formula, we were able to find the lengths of all three sides of the triangle.