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Analytic Geometry: Lines and Circles · Ages 16–17

Distance and Midpoint in Analytic Geometry Worksheet

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Designed for 11th grade math students. Compute the distance, midpoint, or missing coordinate for each question, then verify your results with the step-by-step answers.

  1. Find the distance between the origin (0, 0) and the point (6, 8).

  2. Find the midpoint of the line segment joining (2, 7) and (8, 3).

  3. Calculate the distance between point P(-3, 2) and point Q(1, -1).

  4. The midpoint of segment CD is M(3, 4). If point C is at (1, 2), find the coordinates of point D.

  5. A triangle has vertices at A(1, 1), B(4, 5), and C(1, 5). Compute the lengths of sides AB, BC, and AC.

  6. Using the side lengths from the previous item (3, 4, and 5), classify triangle ABC by its sides and angles.

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Distance and Midpoint in Analytic Geometry Worksheet

Answer key

  1. Find the distance between the origin (0, 0) and the point (6, 8).

    10 units. Calculation: d = sqrt((6 - 0)^2 + (8 - 0)^2) = sqrt(36 + 64) = sqrt(100) = 10.

  2. Find the midpoint of the line segment joining (2, 7) and (8, 3).

    (5, 5). Calculation: M = ((2 + 8)/2, (7 + 3)/2) = (10/2, 10/2) = (5, 5).

  3. Calculate the distance between point P(-3, 2) and point Q(1, -1).

    5 units. Calculation: d = sqrt((1 - (-3))^2 + (-1 - 2)^2) = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = sqrt(25) = 5.

  4. The midpoint of segment CD is M(3, 4). If point C is at (1, 2), find the coordinates of point D.

    (5, 6). Calculation: (1 + xD)/2 = 3 gives xD = 5; (2 + yD)/2 = 4 gives yD = 6.

  5. A triangle has vertices at A(1, 1), B(4, 5), and C(1, 5). Compute the lengths of sides AB, BC, and AC.

    AB = 5, BC = 3, AC = 4. Calculation: AB = sqrt((4 - 1)^2 + (5 - 1)^2) = sqrt(9 + 16) = 5; BC = 4 - 1 = 3; AC = 5 - 1 = 4.

  6. Using the side lengths from the previous item (3, 4, and 5), classify triangle ABC by its sides and angles.

    Scalene and right-angled. All three sides have distinct lengths (3, 4, 5), and 3^2 + 4^2 = 5^2 (9 + 16 = 25), satisfying the Pythagorean theorem.