The distance of any point (x, y) from the y-axis is defined as the absolute value of its x-coordinate. For the point (-2, -3), the x-coordinate is -2. The absolute value of -2 is |-2| = 2. Thus, the distance from the y-axis is 2 units.
72
Azam walks 30 meters North, turns right and walks 40 meters, turns right again and walks 20 meters, and finally turns right and walks 40 meters. What is his final distance from the starting point?
Tracking the movement on a coordinate plane starting at (0,0): he moves to (0,30), then to (40,30), then to (40,10), and finally to (0,10). The final position is (0,10). The distance from the origin (0,0) to (0,10) is calculated using the distance formula, resulting in 10 meters.
73
Determine the centroid of a triangle LMN with vertices at L(1,1,1), M(3,3,3), and N(5,5,5).
The centroid of a triangle in 3D space is calculated by taking the average of the x, y, and z coordinates of its vertices. For vertices (1,1,1), (3,3,3), and (5,5,5), the centroid is ((1+3+5)/3, (1+3+5)/3, (1+3+5)/3), which equals (3,3,3).
74
If the origin (0,0) is the midpoint of the line segment connecting points (a, 3) and (5, b), what are the values of a and b?
The midpoint formula is ((x1+x2)/2, (y1+y2)/2). Setting the midpoint to (0,0) gives (a+5)/2 = 0 and (3+b)/2 = 0. Solving these equations, we find a + 5 = 0, so a = -5, and 3 + b = 0, so b = -3.
75
Given that the point (2, 3) is the midpoint of the line segment connecting (a, 3) and (5, b), find the values of a and b.
The midpoint formula for coordinates (x, y) is ((x1+x2)/2, (y1+y2)/2). Setting the midpoint (2, 3) equal to ((a+5)/2, (3+b)/2), we solve the equations: (a+5)/2 = 2 implies a+5 = 4, so a = -1. Similarly, (3+b)/2 = 3 implies 3+b = 6, so b = 3.
76
Calculate the coordinates of the midpoint of the line segment connecting the points (3, -1) and (-3, 1).
To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2), use the formula ((x1 + x2)/2, (y1 + y2)/2). Substituting the given values: ((3 + -3)/2, (-1 + 1)/2) results in (0/2, 0/2), which simplifies to (0, 0).
77
In which quadrant of the Cartesian plane does the point (-2, -3) reside?
In the Cartesian coordinate system, the plane is divided into four quadrants based on the signs of the x and y coordinates. The first quadrant has positive x and y, the second has negative x and positive y, the third has both negative x and y, and the fourth has positive x and negative y. Since both coordinates are negative, the point lies in the third quadrant.
78
What is the perpendicular distance of a point defined by coordinates (a, b) from the x-axis?
In a Cartesian coordinate system, the x-axis is defined by the equation y = 0. The perpendicular distance from any point (x, y) to the x-axis is given by the absolute value of its y-coordinate. Therefore, for the point (a, b), the distance is |b|. Given the options provided, b is the correct representation.
79
For any point (x, y) located on the y-axis, what must be true regarding its coordinates?
In a standard Cartesian coordinate system, the y-axis is defined as the vertical line where the horizontal distance from the origin is zero. Therefore, for every point lying on the y-axis, the x-coordinate must be zero, regardless of the y-value.
80
If point A is north of point B and point C is west of point B, what is the direction of A relative to C?
Using a coordinate system where B is the origin (0,0), A is at (0,1) because it is North, and C is at (-1,0) because it is West. To find the direction of A relative to C, we look at the vector from C to A, which is (1,1). This vector points into the first quadrant, which corresponds to the North-East direction.
