Bush Lane
09/08/2023 · Middle School
Preguntard 11 coordinate is in the shape of a rectangle. A map of the garden is drawn on a garden are \( (0,-5),(6,3),(2,6) \), and \( (-4,-2) \). What is the area of the garden?
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Step-by-step Solution
Para encontrar el área del jardín representado por los puntos \( (0,-5) \), \( (6,3) \), \( (2,6) \) y \( (-4,-2) \), podemos usar la fórmula del área de un polígono dada por sus vértices:
\[
\text{Área} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\]
Donde los puntos son:
- \( (x_1, y_1) = (0, -5) \)
- \( (x_2, y_2) = (6, 3) \)
- \( (x_3, y_3) = (2, 6) \)
- \( (x_4, y_4) = (-4, -2) \)
Sustituyendo los valores en la fórmula:
\[
\text{Área} = \frac{1}{2} \left| 0 \cdot 3 + 6 \cdot 6 + 2 \cdot (-2) + (-4) \cdot (-5) - (-5 \cdot 6 + 3 \cdot 2 + 6 \cdot (-4) + (-2) \cdot 0) \right|
\]
Calculamos cada término:
1. \( 0 \cdot 3 = 0 \)
2. \( 6 \cdot 6 = 36 \)
3. \( 2 \cdot (-2) = -4 \)
4. \( -4 \cdot (-5) = 20 \)
Sumando estos resultados:
\[
0 + 36 - 4 + 20 = 52
\]
Ahora calculamos la parte de \( y \):
1. \( -5 \cdot 6 = -30 \)
2. \( 3 \cdot 2 = 6 \)
3. \( 6 \cdot (-4) = -24 \)
4. \( -2 \cdot 0 = 0 \)
Sumando estos resultados:
\[
-30 + 6 - 24 + 0 = -48
\]
Ahora sustituimos en la fórmula del área:
\[
\text{Área} = \frac{1}{2} \left| 52 - (-48) \right| = \frac{1}{2} \left| 52 + 48 \right| = \frac{1}{2} \left| 100 \right| = \frac{100}{2} = 50
\]
Por lo tanto, el área del jardín es \( 50 \) unidades cuadradas.
Quick Answer
El área del jardín es 50 unidades cuadradas.
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