Robert Bosch Coordinate Geometry Quiz 2

Ans: (5, 5)
Explanation:
The center of the circle lies at the midpoint of the diameter.
The midpoint of AB = ((4 + 6) / 2, (2 + 8) / 2) = (5, 5)

Answer : y = -0.5x + 3.5
Explanation:
The slope of the line perpendicular to y = 2x - 1 is the negative reciprocal of 2, which is -1/2. So, the equation of line L can be written as y - 3 = (-1/2)(x - 1).
Simplifying, we get y = -0.5x + 3.5, which is option A.

Ans: 5 units
Explanation: The distance between two points (x1, y1) and (x2, y2) is given by the distance formula: √((x2 - x1)^2 + (y2 - y1)^2). Plugging in the given values: √((6 - 3)^2 + (8 - 4)^2) = √(9 + 16) = √25 = 5 units.

Ans: Center: (3, -2), Radius: 5
Explanation: The standard equation of a circle with center (h, k) and radius r is given by (x - h)^{2} + (y - k)^{2} = r^{2}
Comparing the given equation with the standard form,
we have: (x - 3)^{2} + (y + 2)^{2} = 25
Therefore, the center of the circle is (h, k) = (3, -2), and the radius r = √25 = 5.
Therefore, the correct answer is (a) Center: (3, -2), Radius: 5.

Ans: (3, 11.6) and (8, 11.6)
Explanation: The area of a rectangle is given by the product of its length and width. Let the width of the rectangle be "w" and its length be "l".
We are given that the area is 48 square units, so lw = 48.
The coordinates of two vertices are (3, 2) and (8, 2), which means the width of the rectangle is 8 - 3 = 5 units.
Now, we can find the length: lw = 48, w = 5, so l = 48 / 5 = 9.6 units.
Since the rectangle has sides parallel to the x and y axes, the other two vertices will have the same y-coordinate as (3, 2) and (8, 2), and the x-coordinates will differ by the length of the rectangle.
Therefore, the other two vertices are (3, 2 + 9.6) ≈ (3, 11.6) and (8, 2 + 9.6) ≈ (8, 11.6).
Therefore, the correct answer is (b) (3, 11.6) and (8, 11.6).

Ans: Option A
Explanation:
The standard equation of a circle with center (h, k) and radius r is given by
(x - h)^2 + (y - k)^2 = r^2.
Comparing the given equation with the standard form,
we have: (x + 2)^2 + (y - 3)^2 = 25.
Therefore, the center of the circle is (h, k) = (-2, 3), and the radius r = √25 = 5.
Therefore, the correct answer is (a) Center: (-2, 3), Radius: 5.

Ans: (x - 4)^2 + (y - 6)^2 = 18
Explanation: The midpoint of the diameter is the center of the circle.
The midpoint of (1, 3) and (7, 9) = ((1 + 7) / 2, (3 + 9) / 2) = (4, 6).
The radius of the circle is half of the diameter, which is the distance between the endpoints. Radius = √18
Therefore, the equation of the circle is (x - 4)^{2} + (y - 6)^{2} = √18^{2} = 18

Answer : 2√66 square units
Explanation: To find the area of a triangle with side lengths a, b, and c, we can use Heron's formula: Area = √(s(s-a)(s-b)(s-c)),
where s is the semi-perimeter of the triangle, given by s = (a + b + c) / 2.
In this case, the semi-perimeter s = (5 +7 + 10) / 2 = 11.
Therefore, the area = √(11(11-5)(11-7)(11-10))
= √(11*6*4*1) = √(264) ≈ 2√66

Ans: 6 square units
Explanation: To find the x-intercept, we set y = 0 and solve for x: 3x + 4(0) = 12 → x = 4.
To find the y-intercept, we set x = 0 and solve for y: 3(0) + 4y = 12 → y = 3.
The points A and B are (4, 0) and (0, 3), respectively.
The area of the triangle formed by points
A, B, and the origin is given by A = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|, where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three points.
Plugging in the values: A = 1/2 * |4(3 - 0) + 0(0 - 3) + 0(0 - 0)| = 1/2 * |12| = 6 square units.
Therefore, the correct answer is (a) 6 square units.

Answer: 2
Explanation: The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:
Area = 0.5 * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|
Substitute the values from points A, B, and C and set the area equal to 9:
9 = 0.5 * |(2(3 - k) + (-1)(k - 6) + 4(6 - 3))|
Solving for k, we get k = 2, which is option C.