Coordinate Geometry MCQ Test: Trivia!

Explanation
The distance of a point from the x-axis is the perpendicular distance of the point from the x-axis. In this case, the point P (2,3) lies on the y-axis, which is perpendicular to the x-axis. Therefore, the distance of point P from the x-axis is equal to the y-coordinate of the point, which is 3.

Explanation
The given points (-4,0), (4,0), (0,3) form a triangle. To determine the type of triangle, we can calculate the lengths of the three sides. The distance between (-4,0) and (4,0) is 8 units, making it the base of the triangle. The distance between (-4,0) and (0,3) is 5 units, and the distance between (4,0) and (0,3) is also 5 units. Since two sides have the same length, this triangle is an isosceles triangle.

Explanation
The distance between two points in a coordinate plane can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this case, the y-coordinates of points A and B are 6 and -2 respectively. By subtracting the y-coordinates and taking the absolute value, we get the vertical distance between the points, which is 8. Therefore, the correct answer is 8.

Explanation
To find the area of a triangle, we can use the formula: 1/2 * base * height. In this case, the base is the distance between points A and B, which is 7-3 = 4. The height is the perpendicular distance from point C to the line AB. To find this, we can use the formula for the distance between a point and a line. The equation of line AB is y = 0, so the distance from point C (8,4) to this line is 4. Therefore, the area of the triangle is 1/2 * 4 * 4 = 8.

Explanation
The correct answer is 11 units from origin. This can be determined by calculating the distance between point A (-5,6) and the origin (0,0) using the distance formula. The distance formula is given by √((x2-x1)^2 + (y2-y1)^2). Plugging in the values, we get √((-5-0)^2 + (6-0)^2) = √(25 + 36) = √61. Therefore, the distance between point A and the origin is 61 units.

Explanation
The length of the diagonal of a rectangle can be found using the Pythagorean theorem. The theorem states that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the two sides. In this case, the length of one side is 8 units (the difference in y-coordinates between A and O) and the length of the other side is 6 units (the difference in x-coordinates between O and B). Applying the Pythagorean theorem, we have: diagonal^2 = 8^2 + 6^2 = 64 + 36 = 100. Taking the square root of both sides, we find that the length of the diagonal is 10 units.

Explanation
If the points (1,x), (5,2), and (9,5) are collinear, it means that they lie on the same straight line. To determine the value of x, we can use the concept of slope. The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

Using the points (1,x) and (5,2), the slope is (2 - x) / (5 - 1).
Using the points (5,2) and (9,5), the slope is (5 - 2) / (9 - 5).

Since the points are collinear, the two slopes should be equal. Setting the two expressions equal to each other and solving for x, we get (2 - x) / 4 = 3 / 4. Solving this equation gives x = -1.

Explanation
The perimeter of a triangle is the sum of the lengths of its three sides. To find the length of each side, we can use the distance formula. The distance between (0,4) and (0,0) is 4 units (vertical side). The distance between (0,0) and (3,0) is 3 units (horizontal side). The distance between (3,0) and (0,4) is 5 units (diagonal side). Adding these three side lengths together gives us a perimeter of 12 units.

Explanation
The correct answer is (0,b) because when a point lies on the y-axis, the x-coordinate is always 0. In this case, all the other options have non-zero x-coordinates, so they cannot be the correct answer. The only option with x-coordinate 0 is (0,b), which means it lies on the y-axis.

Explanation
To find the radius of the circle, we need to find the distance between the two endpoints of the diameter. We can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by the square root of ((x2 - x1)^2 + (y2 - y1)^2). In this case, the distance between (2, 4) and (-3, -1) is equal to the square root of ((-3 - 2)^2 + (-1 - 4)^2), which simplifies to the square root of (25 + 25) = square root of 50. Therefore, the radius of the circle is the square root of 50.

Explanation
The point (0, 0) lies on the perpendicular bisector of the line segment joining points A (-2, -5) and B (2, 5) because it is equidistant from both A and B. The perpendicular bisector of a line segment is a line that cuts the segment into two equal parts and is perpendicular to the line segment. In this case, the line passing through A and B has a slope of 1, so the perpendicular bisector will have a slope of -1. The midpoint of AB is (0, 0), which also lies on the perpendicular bisector. Therefore, (0, 0) is the correct answer.

Explanation
The ratio in which the x-axis divides the line segment joining the points (5,4) and (2,-3) is 4:3. This means that the x-coordinate of the point where the x-axis intersects the line segment is 4/7 times the distance from (2,-3) to (5,4), and the x-coordinate of the other point on the line segment is 3/7 times the distance.

Explanation
To find the fourth vertex of the parallelogram, we can use the fact that opposite sides of a parallelogram are parallel and equal in length. We can start by finding the length and slope of one side of the parallelogram. Using the given coordinates, the length of side AB is sqrt((6 - (-2))^2 + (7 - 3)^2) = sqrt(64 + 16) = sqrt(80) = 4sqrt(5). The slope of AB is (7 - 3)/(6 - (-2)) = 4/8 = 1/2. Since opposite sides of a parallelogram are parallel, the slope of DC is also 1/2. We can use this slope and the coordinates of C (8, 3) to find the equation of line DC: y - 3 = (1/2)(x - 8). Simplifying this equation, we get y = (1/2)x - 1. Since D is the intersection of lines AB and DC, we can solve the system of equations formed by the equations of AB and DC. Solving the system, we find that the coordinates of D are (0, -1).

Explanation
The point which divides a line segment internally in a given ratio lies in the same quadrant as the point with the larger coordinates. In this case, the point (7, -6) has larger coordinates than the point (3, 4), so the point dividing the line segment in a 1:2 ratio will also be in the I quadrant.

Explanation
If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then AP = 1/2 AB. This can be explained using the concept of section formula. The section formula states that if a point P divides a line segment AB in the ratio m:n, then the coordinates of P can be found using the formula: P = (mx2 + nx1)/(m + n), (my2 + ny1)/(m + n). In this case, m = 1 and n = 2. Substituting the values into the formula, we get P = (2*8 + 1*4)/(1 + 2), (2*4 + 1*2)/(1 + 2) = (16 + 4)/3, (8 + 2)/3 = 20/3, 10/3. Thus, AP = 1/2 AB.