To find the missing side lengths, perimeter, and area of triangle ABC on the coordinate plane, we need the coordinates of the vertices A, B, and C.
For example, let's say you have the following coordinates for the vertices:
- A(0, 0)
- B(4, 0)
- C(2, 3)
Step 1: Calculate the Lengths of the Sides
-
Length AB: \[ AB = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{16} = 4 \]
-
Length BC: \[ BC = \sqrt{(2 - 4)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \]
-
Length AC: \[ AC = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \]
Step 2: Calculate the Perimeter
\[ \text{Perimeter} = AB + BC + AC = 4 + \sqrt{13} + \sqrt{13} = 4 + 2\sqrt{13} \]
Step 3: Calculate the Area
Using the formula for the area of a triangle, where \( b \) is the length of side BC and \( h \) is the height from point A to line BC: \[ A = \frac{bh}{2} \] The base \( BC = \sqrt{13} \) and the height from A(0, 0) is the y-coordinate of point C(2, 3), which is 3: \[ \text{Area} = \frac{\sqrt{13} \cdot 3}{2} = \frac{3\sqrt{13}}{2} \]
Final Answers
- Side Length AB: 4
- Side Length BC: \(\sqrt{13}\)
- Side Length AC: \(\sqrt{13}\)
- Perimeter of Triangle ABC: \(4 + 2\sqrt{13}\)
- Area of Triangle ABC: \(\frac{3\sqrt{13}}{2}\)
Make sure to use the correct coordinates and calculations based on your specific triangle ABC if they differ from the example provided.
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