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Sagot :
Sure, let's break down the solution to find the perimeter and area of the polygon with vertices [tex]\( W(-4,28) \)[/tex], [tex]\( X(8,23) \)[/tex], [tex]\( Y(-13,16) \)[/tex], and [tex]\( Z(-4,11) \)[/tex].
Part A. Perimeter:
To find the perimeter of the polygon, we need to calculate the distances between each pair of consecutive vertices and then sum them up.
1. Calculate distance [tex]\( W \)[/tex] to [tex]\( X \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(X_x - W_x)^2 + (X_y - W_y)^2} \][/tex]
[tex]\( W(-4, 28) \)[/tex] and [tex]\( X(8, 23) \)[/tex]:
[tex]\[ = \sqrt{(8 - (-4))^2 + (23 - 28)^2} = \sqrt{(12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \][/tex]
2. Calculate distance [tex]\( X \)[/tex] to [tex]\( Y \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(Y_x - X_x)^2 + (Y_y - X_y)^2} \][/tex]
[tex]\( X(8, 23) \)[/tex] and [tex]\( Y(-13, 16) \)[/tex]:
[tex]\[ = \sqrt{(-13 - 8)^2 + (16 - 23)^2} = \sqrt{(-21)^2 + (-7)^2} = \sqrt{441 + 49} = \sqrt{490} \approx 22.1 \][/tex]
3. Calculate distance [tex]\( Y \)[/tex] to [tex]\( Z \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(Z_x - Y_x)^2 + (Z_y - Y_y)^2} \][/tex]
[tex]\( Y(-13, 16) \)[/tex] and [tex]\( Z(-4, 11) \)[/tex]:
[tex]\[ = \sqrt{(-4 - (-13))^2 + (11 - 16)^2} = \sqrt{(9)^2 + (-5)^2} = \sqrt{81 + 25} = \sqrt{106} \approx 10.3 \][/tex]
4. Calculate distance [tex]\( Z \)[/tex] to [tex]\( W \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(W_x - Z_x)^2 + (W_y - Z_y)^2} \][/tex]
[tex]\( Z(-4, 11) \)[/tex] and [tex]\( W(-4, 28) \)[/tex]:
[tex]\[ = \sqrt{(-4 - (-4))^2 + (28 - 11)^2} = \sqrt{(0)^2 + (17)^2} = \sqrt{17^2} = 17 \][/tex]
Now, sum up the distances to get the perimeter:
[tex]\[ \text{Perimeter} = 13 + 22.1 + 10.3 + 17 \approx 62.4 \text{ units} \][/tex]
Part B. Area:
To find the area of the polygon, we can use the Shoelace formula (or Gauss's area formula), which is given by:
[tex]\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \][/tex]
For vertices [tex]\( W(-4,28) \)[/tex], [tex]\( X(8,23) \)[/tex], [tex]\( Y(-13,16) \)[/tex], and [tex]\( Z(-4,11) \)[/tex]:
[tex]\[ \begin{aligned} \text{Area} &= \frac{1}{2} \left| (-4 \cdot 23 + 8 \cdot 16 + (-13) \cdot 11 + (-4) \cdot 28) - (28 \cdot 8 + 23 \cdot (-13) + 16 \cdot (-4) + 11 \cdot (-4)) \right| \\ &= \frac{1}{2} \left| ( -92 + 128 - 143 - 112) - ( 224 - 299 - 64 - 44) \right| \\ &= \frac{1}{2} \left| -219 -115 \right| \\ &= \frac{1}{2} \left| -104 \right|\\ &= 52 \end{aligned} \][/tex]
Thus, the area is:
[tex]\[ \text{Area} = 52 \text{ square units} \][/tex]
So, the answers are:
[tex]\[ \begin{aligned} \text{Part A: Perimeter} &= 62.4 \text{ units} \\ \text{Part B: Area} &= 52 \text{ square units} \end{aligned} \][/tex]
Part A. Perimeter:
To find the perimeter of the polygon, we need to calculate the distances between each pair of consecutive vertices and then sum them up.
1. Calculate distance [tex]\( W \)[/tex] to [tex]\( X \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(X_x - W_x)^2 + (X_y - W_y)^2} \][/tex]
[tex]\( W(-4, 28) \)[/tex] and [tex]\( X(8, 23) \)[/tex]:
[tex]\[ = \sqrt{(8 - (-4))^2 + (23 - 28)^2} = \sqrt{(12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \][/tex]
2. Calculate distance [tex]\( X \)[/tex] to [tex]\( Y \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(Y_x - X_x)^2 + (Y_y - X_y)^2} \][/tex]
[tex]\( X(8, 23) \)[/tex] and [tex]\( Y(-13, 16) \)[/tex]:
[tex]\[ = \sqrt{(-13 - 8)^2 + (16 - 23)^2} = \sqrt{(-21)^2 + (-7)^2} = \sqrt{441 + 49} = \sqrt{490} \approx 22.1 \][/tex]
3. Calculate distance [tex]\( Y \)[/tex] to [tex]\( Z \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(Z_x - Y_x)^2 + (Z_y - Y_y)^2} \][/tex]
[tex]\( Y(-13, 16) \)[/tex] and [tex]\( Z(-4, 11) \)[/tex]:
[tex]\[ = \sqrt{(-4 - (-13))^2 + (11 - 16)^2} = \sqrt{(9)^2 + (-5)^2} = \sqrt{81 + 25} = \sqrt{106} \approx 10.3 \][/tex]
4. Calculate distance [tex]\( Z \)[/tex] to [tex]\( W \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(W_x - Z_x)^2 + (W_y - Z_y)^2} \][/tex]
[tex]\( Z(-4, 11) \)[/tex] and [tex]\( W(-4, 28) \)[/tex]:
[tex]\[ = \sqrt{(-4 - (-4))^2 + (28 - 11)^2} = \sqrt{(0)^2 + (17)^2} = \sqrt{17^2} = 17 \][/tex]
Now, sum up the distances to get the perimeter:
[tex]\[ \text{Perimeter} = 13 + 22.1 + 10.3 + 17 \approx 62.4 \text{ units} \][/tex]
Part B. Area:
To find the area of the polygon, we can use the Shoelace formula (or Gauss's area formula), which is given by:
[tex]\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \][/tex]
For vertices [tex]\( W(-4,28) \)[/tex], [tex]\( X(8,23) \)[/tex], [tex]\( Y(-13,16) \)[/tex], and [tex]\( Z(-4,11) \)[/tex]:
[tex]\[ \begin{aligned} \text{Area} &= \frac{1}{2} \left| (-4 \cdot 23 + 8 \cdot 16 + (-13) \cdot 11 + (-4) \cdot 28) - (28 \cdot 8 + 23 \cdot (-13) + 16 \cdot (-4) + 11 \cdot (-4)) \right| \\ &= \frac{1}{2} \left| ( -92 + 128 - 143 - 112) - ( 224 - 299 - 64 - 44) \right| \\ &= \frac{1}{2} \left| -219 -115 \right| \\ &= \frac{1}{2} \left| -104 \right|\\ &= 52 \end{aligned} \][/tex]
Thus, the area is:
[tex]\[ \text{Area} = 52 \text{ square units} \][/tex]
So, the answers are:
[tex]\[ \begin{aligned} \text{Part A: Perimeter} &= 62.4 \text{ units} \\ \text{Part B: Area} &= 52 \text{ square units} \end{aligned} \][/tex]
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