Coordinate Geometry Act Questions

The given points form a parallelogram in the coordinate plane. To find the area of a parallelogram, we can use the formula: base x height. The base of the parallelogram is the distance between the points (0,2) and (8,2), which is 8 units. The height of the parallelogram is the distance between the points (0,2) and (3,6), which is 4 units. Therefore, the area of the parallelogram is 8 x 4 = 32 square units.

The approximate distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the coordinates of the two points are (4, -3) and (-6, 5). Plugging these values into the distance formula, we get:

d = √((-6 - 4)^2 + (5 - (-3))^2)
= √((-10)^2 + (8)^2)
= √(100 + 64)
= √(164)
≈ 12.81

Therefore, the approximate distance between the two points is 12.81.

The slope of a line parallel to another line will have the same slope. In this case, the given line has a slope of 2/5. Therefore, the slope of a line parallel to it will also be 2/5.

The slope of a line can be found by calculating the change in y-coordinates divided by the change in x-coordinates between two points on the line. In this case, the change in y-coordinates is 2 - (-1) = 3, and the change in x-coordinates is 16 - (-2) = 18. Therefore, the slope of the accessibility ramp is 3/18, which simplifies to 1/6.

The slope of a line can be determined by rearranging the equation into slope-intercept form, y = mx + b, where m represents the slope. In this case, rearranging 2x - 3y = 6 gives -3y = -2x + 6, then dividing by -3 yields y = (2/3)x - 2. Therefore, the slope of the line is 2/3.

To find the area of triangle GHJ, we need to calculate the base and height of the triangle. The base is the distance between points G and H, which can be found using the distance formula. The height is the distance between the line containing points G and H and point J, which can be found by calculating the perpendicular distance from point J to the line containing points G and H. Once we have the base and height, we can use the formula for the area of a triangle (1/2 * base * height) to find the area.

In the figure, it is given that OS = ST, which means that the length of OS is equal to the length of ST. The coordinates of T are given as (k, 5). Since OS = ST, the y-coordinate of T should also be 5. Therefore, the value of k should be such that the point T has a y-coordinate of 5. Looking at the answer options, the only value that satisfies this condition is k = 5.

The slope of a line segment can be determined by finding the change in the y-coordinates divided by the change in the x-coordinates. In this case, the line segment AC goes from point A to point C, which has a change in y-coordinate of 2 and a change in x-coordinate of 2. Therefore, the slope of the line segment AC is 2/2 = 1.

Since point B is the midpoint of AC, the coordinates of point C can be found by using the midpoint formula. The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints. In this case, the x-coordinate of point C is (6 + x) / 2 = 5, which gives x = 4. Similarly, the y-coordinate of point C is (7 + y) / 2 = 6, which gives y = 5. Therefore, the coordinates of point C are (4, 5).

The given equation is in the form of Ax + By + C = 0. To convert it to slope-intercept form (y = mx + b), we need to isolate y. By rearranging the equation, we get y = 10x - 8. Therefore, the correct answer is y = 10x - 8.

The distance between two points in a standard (x, y) coordinate plane can be calculated using the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and the y-coordinates of the two points. In this case, the x-coordinate difference is 5-1=4 and the y-coordinate difference is 5-0=5. Therefore, the distance is the square root of (4^2 + 5^2) = square root of (16 + 25) = square root of 41.

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The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. The given equation can be rewritten in slope-intercept form as y = (-5/8)x + 17/8. Therefore, the slope of the original line is -5/8. The negative reciprocal of -5/8 is 8/5, which is the slope of the line perpendicular to the given line.

The distance on the y-axis from point A to point B is 5 units. This can be determined by subtracting the y-coordinate of point A from the y-coordinate of point B, which gives a difference of 5.

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