Geometry: Coordinate Geometry Worksheet

Question 1

Find the distance between A(2, -3) and B(8, 5).

Accepted Answer

The distance is 10 units. Using the distance formula, d = sqrt((8 - 2)^2 + (5 - (-3))^2) = sqrt(6^2 + 8^2) = sqrt(100) = 10.

Question 2

Find the midpoint of the segment with endpoints P(-4, 7) and Q(6, -1).

Accepted Answer

The midpoint is (1, 3). The x-coordinate is (-4 + 6) / 2 = 1, and the y-coordinate is (7 + (-1)) / 2 = 3.

Question 3

Find the slope of the line passing through C(-2, 4) and D(3, -6).

Accepted Answer

The slope is -2. The change in y is -6 - 4 = -10, and the change in x is 3 - (-2) = 5, so the slope is -10 / 5 = -2.

Question 4

Write the equation of the line with slope 3 that passes through the point (2, -5). Give your answer in slope-intercept form.

Accepted Answer

The equation is y = 3x - 11. Substituting (2, -5) into y = 3x + b gives -5 = 6 + b, so b = -11.

Question 5

Determine whether the line through A(1, 2) and B(5, 10) is parallel to the line through C(-3, 4) and D(2, 14). Explain your answer.

Accepted Answer

The lines are parallel because they have the same slope. The slope of AB is (10 - 2) / (5 - 1) = 8 / 4 = 2, and the slope of CD is (14 - 4) / (2 - (-3)) = 10 / 5 = 2.

Question 6

Determine whether the line through E(-1, 3) and F(5, 6) is perpendicular to the line through G(2, -4) and H(5, -10). Explain your answer.

Accepted Answer

The lines are perpendicular. The slope of EF is (6 - 3) / (5 - (-1)) = 3 / 6 = 1/2, and the slope of GH is (-10 - (-4)) / (5 - 2) = -6 / 3 = -2. Since 1/2 and -2 are negative reciprocals, the lines are perpendicular.

Question 7

A triangle has vertices A(0, 0), B(6, 0), and C(6, 8). Find the lengths of all three sides and classify the triangle by its side lengths.

Accepted Answer

The side lengths are AB = 6, BC = 8, and AC = 10. Since all three side lengths are different, the triangle is scalene. It is also a right triangle because 6^2 + 8^2 = 10^2.

Question 8

Find the coordinates of the point that divides the segment from A(2, 1) to B(8, 13) in a 1:2 ratio, measured from A to B.

Accepted Answer

The point is (4, 5). A 1:2 ratio from A to B means the point is one third of the way from A to B. The change from A to B is (6, 12), and one third of that is (2, 4). Adding this to A gives (4, 5).

Question 9

Write the equation of the perpendicular bisector of the segment with endpoints M(-2, 4) and N(4, -2).

Accepted Answer

The equation is y = x. The midpoint of MN is (1, 1). The slope of MN is (-2 - 4) / (4 - (-2)) = -6 / 6 = -1, so the perpendicular slope is 1. A line with slope 1 through (1, 1) is y - 1 = 1(x - 1), which simplifies to y = x.

Question 10

A quadrilateral has vertices A(-3, 1), B(1, 4), C(5, 1), and D(1, -2). Use slopes or distances to determine what type of quadrilateral it is.

Accepted Answer

The quadrilateral is a rhombus. All four side lengths are 5 units: AB, BC, CD, and DA each have distance sqrt(4^2 + 3^2) = 5. Opposite sides are parallel because their slopes match in pairs.

Question 11

Find the area of the triangle with vertices A(1, 1), B(7, 1), and C(4, 6).

Accepted Answer

The area is 15 square units. The base from A to B is 6 units because it is horizontal, and the height from C to the line y = 1 is 5 units. The area is (1/2)(6)(5) = 15.

Question 12

The equation of a circle is (x - 3)^2 + (y + 2)^2 = 49. Identify the center and radius.

Accepted Answer

The center is (3, -2), and the radius is 7. The standard form is (x - h)^2 + (y - k)^2 = r^2, so h = 3, k = -2, and r = sqrt(49) = 7.

Question 13

Write the equation of the circle with center (-4, 5) that passes through the point (2, 13).

Accepted Answer

The equation is (x + 4)^2 + (y - 5)^2 = 100. The radius is the distance from (-4, 5) to (2, 13), which is sqrt(6^2 + 8^2) = 10, so r^2 = 100.

Question 14

A line has equation 2x - 3y = 12. Find its x-intercept and y-intercept.

Accepted Answer

The x-intercept is (6, 0), and the y-intercept is (0, -4). To find the x-intercept, set y = 0 and solve 2x = 12. To find the y-intercept, set x = 0 and solve -3y = 12.

Question 15

A rectangle has vertices A(-2, -1), B(4, -1), C(4, 3), and D(-2, 3). Find the perimeter and area of the rectangle.

Accepted Answer

The perimeter is 20 units, and the area is 24 square units. The width is 4 - (-2) = 6 units, and the height is 3 - (-1) = 4 units. The perimeter is 2(6 + 4) = 20, and the area is 6 times 4 = 24.

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