Can You Pass The Coordinate Geometry Quiz?

Explanation
The midpoint formula states that the midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the endpoints. In this case, the x-coordinate of the midpoint is 4, which is the average of x and 5. Therefore, x = 3. The y-coordinate of the midpoint is 1, which is the average of -2 and y. Therefore, -2 + y = 2, and solving for y gives y = 4.

Explanation
The correct answer is y=-5 because a line parallel to the x-axis has a constant y-value. Since the line passes through the point (-3,-5), the y-value for any point on the line will always be -5. Therefore, the equation of the line is y=-5.

Explanation
Since QP is parallel to the x-axis, the y-coordinate of Q will be the same as the y-coordinate of P, which is 4. We can substitute this value into the equation of RQ to find the x-coordinate of Q. By rearranging the equation, we get 2x = y + 1, so 2x = 4 + 1, and x = 2.5. Therefore, the coordinates of Q are (2.5, 4). To find the distance between P and Q, we can use the distance formula: √((x2 - x1)^2 + (y2 - y1)^2). Plugging in the coordinates, we get √((2.5 - 8)^2 + (4 - 4)^2) = √((-5.5)^2 + 0^2) = √(30.25) = 5.5. Therefore, the distance from P to Q is 5.5.

Explanation
The equation of a straight line passing through a point (3,5) perpendicular to another line can be found by taking the negative reciprocal of the slope of the given line. The given line, 3x+y=6, can be rewritten as y=-3x+6, which has a slope of -3. The negative reciprocal of -3 is 1/3. Using the point-slope form of a line, y-y1=m(x-x1), where (x1,y1) is the given point and m is the slope, we can substitute (3,5) and 1/3 into the equation. Simplifying, we get 3y-x-12=0, which matches the given answer.

Explanation
The midpoint of a line segment is the average of the coordinates of its endpoints. Given that the midpoint is (-2,4) and one endpoint is (2,-2), we can find the coordinates of the other endpoint by subtracting the coordinates of the midpoint from twice the coordinates of one endpoint. Using this formula, we get Q = 2P - (-2,4) = (2*2 - (-2), 2*(-2) - 4) = (-6,10). Therefore, the coordinates of Q are (-6,10).

Explanation
When a line is reflected in another line, the equation of the reflected line is obtained by replacing x with y and y with x in the original equation. In this case, the original equation is y=2x+3. Replacing x with y and y with x, we get x=2y+3. Simplifying this equation, we get x-2y-3=0. However, none of the given options match this equation exactly. The closest option is x-2y+0, which can be simplified to x-2y=0. Therefore, x-2y+0 is the closest approximation to the equation of the reflected line.

Explanation
The x-coordinate of the foot of the perpendicular from the point (1,9) to the line y=x can be found by finding the equation of the line perpendicular to y=x passing through the point (1,9) and then finding the x-coordinate of the intersection point of the two lines. The equation of the line perpendicular to y=x can be found by taking the negative reciprocal of the slope of y=x, which is -1. Using the point-slope form of a line, the equation of the perpendicular line is y - 9 = -1(x - 1). Simplifying this equation gives y = -x + 10. To find the x-coordinate of the intersection point, we set y = x and solve for x. Thus, x = 5.

Explanation
Given three points (-1,6), (0,0), and (3,1), we can determine the fourth vertex of the parallelogram by finding the midpoint of the line segment connecting the first two points and then extending that line segment by the same length to find the fourth point. Similarly, we can find the midpoint of the line segment connecting the second and third points and extend it to find the fourth point. Finally, we can find the midpoint of the line segment connecting the third and first points and extend it to find the fourth point. Therefore, there are three possible positions for the fourth vertex of the parallelogram.

Explanation
The equation of a straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. In this case, the gradient is -3/2, which means that for every 2 units the line moves horizontally (x-axis), it moves downward 3 units (y-axis).

Since the line cuts the x-axis at the point (4,0), we know that when x = 4, y = 0. Plugging these values into the equation y = mx + c, we get 0 = (-3/2)(4) + c. Simplifying this equation gives us c = 6.

Therefore, the equation of the line is 2y = -3x + 12.

Explanation
When a point is reflected in the line x=0, the x-coordinate remains the same, but the sign of the y-coordinate is reversed. So, the new point after the first reflection would be (x,-y).

When this new point is reflected in the line y=0, the y-coordinate remains the same, but the sign of the x-coordinate is reversed. So, the final image point would be (-x, -y).