To find the perimeter of the parallelogram with given vertices, we first need to identify the pairs of adjacent vertices and calculate the lengths of the sides.
The vertices are:
- \( A(0, 0) \)
- \( B(2, 3) \)
- \( C(5, 2) \)
- \( D(3, -1) \)
We'll consider the sides \( AB \), \( BC \), \( CD \), and \( DA \) where:
- \( AB \) connects points \( A \) and \( B \),
- \( BC \) connects points \( B \) and \( C \),
- \( CD \) connects points \( C \) and \( D \),
- \( DA \) connects points \( D \) and \( A \).
Now, we will calculate the lengths of these sides using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length \( AB \):
- \( A(0, 0) \) and \( B(2, 3) \): \[ AB = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 \]
Length \( BC \):
- \( B(2, 3) \) and \( C(5, 2) \): \[ BC = \sqrt{(5 - 2)^2 + (2 - 3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.1623 \]
Length \( CD \):
- \( C(5, 2) \) and \( D(3, -1) \): \[ CD = \sqrt{(3 - 5)^2 + (-1 - 2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 \]
Length \( DA \):
- \( D(3, -1) \) and \( A(0, 0) \): \[ DA = \sqrt{(0 - 3)^2 + (0 - (-1))^2} = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.1623 \]
Total Perimeter:
Now we sum the lengths of all the sides: \[ \text{Perimeter} = AB + BC + CD + DA \approx \sqrt{13} + \sqrt{10} + \sqrt{13} + \sqrt{10} \] This simplifies to: \[ \text{Perimeter} = 2\sqrt{13} + 2\sqrt{10} \approx 2 \cdot 3.6056 + 2 \cdot 3.1623 \] Calculating further: \[ \text{Perimeter} \approx 7.2112 + 6.3246 \approx 13.5358 \]
Rounding this to the nearest tenth, we get: \[ \boxed{13.5} \]
1 answer