Question

finding area draw the figure in a coordinate plane and find its area.

1. square defined by w(5, 0), x(0, 5), y(-5, 0), and z(0, -5)

Explanation:

Step1: Identify the figure type

The given points are \( W(5,0) \), \( X(0,5) \), \( Y(-5,0) \), \( Z(0,-5) \). Let's find the distances between consecutive points to confirm it's a square.

First, distance between \( W(5,0) \) and \( X(0,5) \):
Using distance formula \( d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \)
\( d_{WX}=\sqrt{(0 - 5)^2+(5 - 0)^2}=\sqrt{(-5)^2+5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} \)

Distance between \( X(0,5) \) and \( Y(-5,0) \):
\( d_{XY}=\sqrt{(-5 - 0)^2+(0 - 5)^2}=\sqrt{(-5)^2+(-5)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} \)

Distance between \( Y(-5,0) \) and \( Z(0,-5) \):
\( d_{YZ}=\sqrt{(0 - (-5))^2+(-5 - 0)^2}=\sqrt{(5)^2+(-5)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} \)

Distance between \( Z(0,-5) \) and \( W(5,0) \):
\( d_{ZW}=\sqrt{(5 - 0)^2+(0 - (-5))^2}=\sqrt{(5)^2+(5)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} \)

Now, check the diagonals. Diagonal \( WY \): from \( (5,0) \) to \( (-5,0) \), length is \( |5 - (-5)| = 10 \) (since y - coordinates are same, distance is difference in x - coordinates). Diagonal \( XZ \): from \( (0,5) \) to \( (0,-5) \), length is \( |5-(-5)| = 10 \) (since x - coordinates are same, distance is difference in y - coordinates).

For a square, diagonals are equal and the area of a square in terms of diagonal \( d \) is \( A=\frac{d^2}{2} \)

Step2: Calculate the area

We found that the length of the diagonal \( d = 10 \) (either \( WY \) or \( XZ \))
Using the formula for area of square with diagonal \( d \): \( A=\frac{d^2}{2} \)
Substitute \( d = 10 \) into the formula: \( A=\frac{10^2}{2}=\frac{100}{2}=50 \)

Alternatively, we can plot the points. \( W(5,0) \) is on the x - axis, \( X(0,5) \) is on the y - axis, \( Y(-5,0) \) is on the x - axis (left of origin), \( Z(0,-5) \) is on the y - axis (below origin). The figure is a square centered at the origin with diagonals along the x - axis and y - axis. The length of each diagonal is 10 (from - 5 to 5 on x - axis and from - 5 to 5 on y - axis). The area of a square is also given by \( \frac{1}{2}\times d_1\times d_2 \), where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Since \( d_1 = d_2=10 \), \( A=\frac{1}{2}\times10\times10 = 50 \)

Answer:

The area of the square is \( 50 \) square units.