Robert Bosch Coordinate Geometry Quiz 1

Solution : 5 units
Explanation :
To find the distance between two points (x1, y1) and (x2, y2) in the Cartesian plane, we use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substitute the values: Distance = √((5 - 2)^2 + (7 - 3)^2) Distance = √(3^2 + 4^2) Distance = √(9 + 16) Distance = √25 Distance = 5 units

Solution : (6, 6)
Explanation :
To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2), we use the midpoint formula:
Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2)
Substitute the values:
Midpoint (M) = ((3 + 9) / 2, (4 + 8) / 2)
=> Midpoint (M) = (12 / 2, 12 / 2)
=> Midpoint (M) = (6, 6)

Solution : y = x + 1
Explanation :
To find the equation of a line passing through two points (x1, y1) and (x2, y2), we use the point-slope form of the equation:
y - y1 = m(x - x1)
where m is the slope of the line.
First, find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) m = (5 - 2) / (4 - 1) = 3 / 3 = 1
Now, use one of the given points (1, 2) to find the equation:
y - 2 = 1(x - 1)
=> y - 2 = x - 1
=> y = x + 1
The equation of the line is y = x + 1.

Solution : y = 3x - 10
Explanation :
To find the equation of the perpendicular bisector of a line segment, we need to follow these steps:
Find the midpoint of the line segment.
Midpoint (M) = ((2 + 8) / 2, (6 + 4) / 2) = (5, 5)
Find the slope of the line segment using the formula:
m = (y2 - y1) / (x2 - x1) m = (4 - 6) / (8 - 2) = -2 / 6 = -1/3
The slope of the perpendicular bisector will be the negative reciprocal of the slope of the line segment.
Slope of the perpendicular bisector = -1 / (-1/3) = 3
Now, use the point-slope form with the midpoint (5, 5) and the calculated slope (3) to find the equation:
y - 5 = 3(x - 5)
=> y - 5 = 3x - 15
=> y = 3x - 10
The equation of the perpendicular bisector is y = 3x - 10.

Solution : 6.5 square units
Explanation :
To find the area of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3), we use the formula for the area of a triangle given its coordinates:
Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|
Substitute the values:
Area = |(1(6 - 3) + 4(3 - 2) + 7(2 - 6)) / 2|
=> Area = |(3 + 4 + (-20)) / 2|
=> Area = |-13 / 2|
=> Area = 6.5 square units

Solution: y = -5
Explanation :
A line parallel to the x-axis has a constant y-coordinate. Since it passes through the point (3, -5), the equation will be of the form y = constant.
Since the line is parallel to the x-axis, the constant y-coordinate will be the y-coordinate of the given point.
The equation of the line is y = -5.

Solution : x = -2
Explanation :
A line perpendicular to the y-axis is parallel to the x-axis.
Hence, its equation will be of the form x = constant.
Since the line passes through the point (-2, 5), the constant x-coordinate will be the x-coordinate of the given point.
The equation of the line is x = -2.

Solution : (-4, 7)
Explanation :
To find the reflection of a point (x, y) over the y-axis, we negate the x-coordinate and keep the y-coordinate the same.
The reflection of P(4, 7) over the y-axis is (-4, 7).

Solution : (x - 2)^2 + (y + 3)^2 = 25
Explanation :
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
where h and k are centre points of circle.
Substitute the values:
(x - 2)^2 + (y - (-3))^2 = 5^2
=> (x - 2)^2 + (y + 3)^2 = 25

Solution : -3/2
Explanation :
To find the slope of a line passing through two points (x1, y1) and (x2, y2), we use the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Substitute the values:
Slope (m) = (-4 - 2) / (1 - (-3))
=> Slope (m) = (-6) / (4)
=> Slope (m) = -3/2