Geometry: Coordinate Geometry
Using coordinates to analyze distance, slope, midpoint, and equations of lines
Geometry: Coordinate Geometry
Using coordinates to analyze distance, slope, midpoint, and equations of lines
Geometry - Grade 9-12
- 1
Find the distance between A(2, -3) and B(8, 5).
Use the distance formula with the changes in x and y.
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. - 2
Find the midpoint of the segment with endpoints P(-4, 7) and Q(6, -1).
The midpoint is (1, 3). The x-coordinate is (-4 + 6) / 2 = 1, and the y-coordinate is (7 + (-1)) / 2 = 3. - 3
Find the slope of the line passing through C(-2, 4) and D(3, -6).
Slope equals change in y divided by change in x.
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. - 4
Write the equation of the line with slope 3 that passes through the point (2, -5). Give your answer in slope-intercept form.
The equation is y = 3x - 11. Substituting (2, -5) into y = 3x + b gives -5 = 6 + b, so b = -11. - 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.
Parallel nonvertical lines have equal slopes.
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. - 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.
Perpendicular nonvertical lines have slopes whose product is -1.
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. - 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.
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. - 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.
A 1:2 ratio means the whole segment is split into 3 equal parts.
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). - 9
Write the equation of the perpendicular bisector of the segment with endpoints M(-2, 4) and N(4, -2).
Find the midpoint first, then use the negative reciprocal of the segment's slope.
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. - 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.
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. - 11
Find the area of the triangle with vertices A(1, 1), B(7, 1), and C(4, 6).
Use the horizontal base AB and find the vertical height from C to that base.
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. - 12
The equation of a circle is (x - 3)^2 + (y + 2)^2 = 49. Identify the center and radius.
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. - 13
Write the equation of the circle with center (-4, 5) that passes through the point (2, 13).
Find the radius using the distance formula, then use the standard circle equation.
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. - 14
A line has equation 2x - 3y = 12. Find its x-intercept and y-intercept.
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. - 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.
Because the sides are horizontal and vertical, subtract x-values for width and y-values for height.
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.