Question

the points then click the \graph quadrilateral\ button. then find the area of the parallelogram. click on the graph to plot a point. click a point to delete it.

Explanation:

Step1: Identify base and height

First, we need to find the length of the base and the height of the parallelogram. Let's assume the coordinates of the points: From the graph, let's find the horizontal and vertical distances. Let's take side EF or EH. Wait, actually, for a parallelogram, area is base × height. Let's find the length of the base (horizontal or vertical) and the height (perpendicular distance). Let's look at the coordinates. Let's assume point E is at (-1, 0), F at (-1, -3), G at (6, -5), H at (6, -2)? Wait, maybe better to find the length of the base as the horizontal distance between two points, and the height as the vertical distance, or vice versa. Wait, actually, looking at the graph, the vertical side (from E to F) has length: from y=0 to y=-3, so length 3? Wait, no, maybe the base is the horizontal distance between E (-1,0) and H (6,-2)? No, better to use the formula for area of parallelogram: base × height, where base is the length of one side, and height is the perpendicular distance between that side and the opposite side.

Wait, let's find the coordinates properly. Let's assume:

Point E: (-1, 0)

Point F: (-1, -3) (since it's directly below E, same x-coordinate, y from 0 to -3, so length 3? Wait, no, from E (-1,0) to F (-1,-3) is vertical, length 3. Then the horizontal distance between E (-1,0) and H (6,-2)? Wait, no, maybe the base is the horizontal length. Wait, actually, looking at the graph, the horizontal distance between E (-1,0) and H (6,-2) – no, maybe the base is the length of EF, but EF is vertical. Wait, no, in a parallelogram, opposite sides are equal and parallel. So EF and HG are vertical, EH and FG are horizontal? Wait, no, let's check the coordinates. Let's say E is (-1, 0), F is (-1, -3), G is (6, -5), H is (6, -2). Then EF is from (-1,0) to (-1,-3): length is |0 - (-3)| = 3. The horizontal distance between E (-1,0) and H (6,-2) – no, the horizontal component between E and H: x from -1 to 6, so 7 units? Wait, no, the base should be the length of EH or FG, and the height is the vertical distance between the horizontal sides. Wait, maybe I made a mistake. Let's re-express:

Wait, the area of a parallelogram is base × height, where base is the length of one side, and height is the perpendicular distance between that side and the opposite side. Let's find the length of the base (horizontal) and the height (vertical). Let's take the base as the distance between E (-1, 0) and H (6, -2)? No, that's not horizontal. Wait, maybe the vertical side is EF: from (-1,0) to (-1,-3), length 3. Then the horizontal distance between E (-1,0) and H (6,-2) – no, the horizontal component is 7 (from x=-1 to x=6 is 7 units). Wait, no, the height would be the horizontal distance if the base is vertical. Wait, actually, if the base is vertical (length 3), then the height is the horizontal distance between the two vertical sides. The vertical sides are x=-1 (EF) and x=6 (HG). So the horizontal distance between x=-1 and x=6 is |6 - (-1)| = 7. Then area is base × height = 3 × 7 = 21? Wait, no, that doesn't seem right. Wait, maybe the base is horizontal. Let's check the horizontal side: from E (-1,0) to H (6,-2) – no, that's not horizontal. Wait, maybe the coordinates are E (-1, 0), F (-1, -3), G (5, -5), H (5, -2)? Wait, the graph shows H at (6, -2) and G at (6, -5)? Wait, the grid is 1 unit per square. Let's count the squares. From E (-1,0) to F (-1,-3): that's 3 units down (vertical). From F (-1,-3) to G (6,-5): that's 7 units right (from x=-1 to x=6 is 7) and 2 units down. From G (6,-5) to H (6,-2): 3 units up (vertical). From H (6,-2) to E (-1,0): 7 units left and 2 units up. So the vertical sides (EF and HG) have length 3 (from y=0 to y=-3, and y=-2 to y=-5: both length 3). The horizontal distance between the vertical sides (x=-1 and x=6) is 7 units (since 6 - (-1) = 7). Therefore, the area of the parallelogram is base (length of vertical side) × height (horizontal distance between vertical sides) = 3 × 7 = 21. Wait, but let's confirm with another method. The area can also be calculated as the magnitude of the cross product of two adjacent sides. Let's take vectors EH and EF. Vector EH: from E (-1,0) to H (6,-2): (6 - (-1), -2 - 0) = (7, -2). Vector EF: from E (-1,0) to F (-1,-3): (0, -3). The cross product in 2D is (7)(-3) - (-2)(0) = -21 - 0 = -21. The magnitude is 21, so area is 21.

Step2: Calculate area

Using the formula for the area of a parallelogram: \( \text{Area} = \text{base} \times \text{height} \). Here, the base (length of vertical side) is 3 (from y=0 to y=-3, so 3 units), and the height (horizontal distance between the vertical sides) is 7 (from x=-1 to x=6, so 7 units). Thus, \( \text{Area} = 3 \times 7 = 21 \).

Answer:

21