To find the perimeter and area of the triangular stained glass window with vertices at the points \((-2, 1)\), \((-6, -3)\), and \( (2, -3)\), we'll follow these steps:
Step 1: Calculate the lengths of the sides using the distance formula.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Side between \((-2, 1)\) and \((-6, -3)\):
\[ d_1 = \sqrt{((-6) - (-2))^2 + ((-3) - 1)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.7 \text{ ft} \]
Side between \((-6, -3)\) and \((2, -3)\):
\[ d_2 = \sqrt{(2 - (-6))^2 + ((-3) - (-3))^2} = \sqrt{(8)^2 + (0)^2} = \sqrt{64} = 8 \text{ ft} \]
Side between \((2, -3)\) and \((-2, 1)\):
\[ d_3 = \sqrt{((-2) - 2)^2 + (1 - (-3))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.7 \text{ ft} \]
Step 2: Calculate the perimeter.
The perimeter \(P\) of the triangle is the sum of the lengths of all sides: \[ P = d_1 + d_2 + d_3 \approx 5.7 + 8 + 5.7 = 19.4 \text{ ft} \]
Step 3: Calculate the area using the formula for the area of a triangle given its vertices.
The area \(A\) of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the following formula: \[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the vertex coordinates \((-2, 1)\), \((-6, -3)\), and \((2, -3)\): \[ A = \frac{1}{2} \left| -2(-3 - (-3)) + (-6)(-3 - 1) + 2(1 - (-3)) \right| \]
This simplifies to: \[ A = \frac{1}{2} \left| -2(0) + (-6)(-4) + 2(4) \right| = \frac{1}{2} \left| 0 + 24 + 8 \right| = \frac{1}{2} \left| 32 \right| = 16 \text{ ft}^2 \]
Final Results
Now, we can fill in the table with the computed values:
| Measure | Value | |---------|------------| | perimeter | 19.4 ft | | area | 16.0 ft² |
Thus, the perimeter of the triangular stained glass window is approximately 19.4 ft, and the area is 16.0 ft².
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