Question

find the area of the polygon with the given vertices. j(-3, 4), k(4, 4), l(3, -3) the area is square units.

Explanation:

Step1: Identify the shape

The polygon with vertices \( J(-3, 4) \), \( K(4, 4) \), \( L(3, -3) \) is a triangle. We can use the formula for the area of a triangle given coordinates, or we can use the base and height. First, find the length of the base \( JK \). Since \( J \) and \( K \) have the same \( y \)-coordinate (4), the length of \( JK \) is the difference in the \( x \)-coordinates.
\( \text{Length of } JK = |4 - (-3)| = |7| = 7 \)

Step2: Find the height

The height is the vertical distance from point \( L \) to the line \( JK \). The line \( JK \) is horizontal (since \( y = 4 \) for both \( J \) and \( K \)), so the height is the absolute difference between the \( y \)-coordinate of \( L \) and 4.
\( \text{Height} = |4 - (-3)| = |7| = 7 \)? Wait, no. Wait, the \( y \)-coordinate of \( L \) is -3, so the vertical distance from \( L \) to \( y = 4 \) is \( |4 - (-3)| = 7 \)? Wait, no, actually, the base is \( JK \), which is horizontal, so the height is the vertical distance from \( L \) to the line \( JK \). Wait, maybe a better way is to use the formula for the area of a triangle with coordinates: \( \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \)

Let's use that formula. Let \( (x_1, y_1) = (-3, 4) \), \( (x_2, y_2) = (4, 4) \), \( (x_3, y_3) = (3, -3) \)

Step3: Apply the formula

\( \text{Area} = \frac{1}{2} |(-3)(4 - (-3)) + 4((-3) - 4) + 3(4 - 4)| \)
\( = \frac{1}{2} |(-3)(7) + 4(-7) + 3(0)| \)
\( = \frac{1}{2} |-21 - 28 + 0| \)
\( = \frac{1}{2} |-49| \)
\( = \frac{1}{2} \times 49 = 24.5 \)

Wait, alternatively, let's check the base and height again. The base \( JK \) is from \( x = -3 \) to \( x = 4 \), so length 7. The height is the vertical distance from \( L(3, -3) \) to the line \( y = 4 \), which is \( 4 - (-3) = 7 \)? Wait, no, the height of a triangle is the perpendicular distance from the base to the opposite vertex. Since \( JK \) is horizontal, the perpendicular distance is vertical, so the \( y \)-coordinate difference. But wait, the base is \( JK \), and the third point is \( L \). Wait, maybe I made a mistake in the base. Wait, actually, the base is \( JK \), length 7, and the height is the vertical distance from \( L \) to \( JK \), but actually, the height should be the vertical distance, but let's plot the points: \( J(-3,4) \), \( K(4,4) \), \( L(3,-3) \). So \( JK \) is a horizontal line segment. The height is the vertical distance from \( L \) to \( JK \), which is \( 4 - (-3) = 7 \)? But then the area would be \( \frac{1}{2} \times 7 \times 7 = 24.5 \), which matches the formula. So that's correct.

Answer:

24.5