On the design plan for a garden, a straight path runs from (-25, 20) to (40, 36). A lamp is going to be placed at the midpoint of the path. Determine the coordinates for the lamp.

\triangle ABC has vertices at A(-4, 4), B(-4, -2), and C(2, -2).

a. Determine the equation of the median from B to AC.

b. Is the median for part a) also an altitude? Explain how you know.

\triangle LMN has vertices at L(O, 4), M(-5, 2), and N(2, -2). Determine the equation of the perpendicular bisector that passes through MN.

Which point is closer to the origin: P(-24, 56) or Q(35, -43)?

A builder needs to connect a partially built house to a temporary power supply. On the plan, the coordinates of the house are (20, 110) and the coordinates of the power supply are (105, 82). What is the least amount of cable needed?

\triangle QRS has vertices at Q(2, 6), R(—3, 1), and 5(6, 2). Determine the perimeter of the triangle.

\triangle XYZ as vertices at X(1, 6), Y(-3, 2), and Z(9, 4). Determine the length of the longest median in the triangle.

a. Determine the equation of the circle that is centred at (0, 0) and passes through point (-8, 15).

b. Identify the coordinates of the intercepts and three other points on the circle.

A circle has a diameter with endpoints C(20, -21) and D(-20, 21). Determine the equation of the circle.

Determine the equation of this circle.

The point (-2, k) lies on the circle x^2 + y^2 = 20. Determine the values of k. Show all the steps in your solution.

\triangle ABC has vertices as shown. Use analytic geometry to categorize \triangle ABC. Show your work.

A triangle has vertices at A(1, 1), B(-2, -1), and C(3, -2). Calculate the side lengths to determine whether the triangle is isosceles, equilateral, or scalene.

Show that the quadrilateral with vertices at J(-1, 1), K6, 4), L(8, 4), and M(4, 1) is a rhombus. What is the length of the side?

Determine the type of quadrilateral described by the vertices R(-3, 2), 5(- 1, 6), T(3, 5), and U(1, 1). Show all the steps in your solution.

A quadrilateral has vertices at A(-3, 1), B(-5, -9), C(7, -1), and D(3, 3). The mid segments of the quadrilateral form ?

Which of the points A(10, 10),B (-7, 3), and C(0, -14) lie on a circle with centre O(5, -2)? What is the radius of this circle?

A triangle has vertices at P(-2, 7), Q(-4, 2), and R(6, -2).

a. Categorize \triangle PQR .

b. Show that the mid point of the hypotenuse is the same distance from each vertex. What is the distance?

Show that points (6, 7) and (-9, 2) are the endpoints of a chord in a circle with centre (0, 0).

Quadrilateral JKZM has vertices as shown. Show that the diagonals of the quadrilateral bisect each other by showing they share the midpoint. What is the mid point?

\triangle PQR has vertices at P(0, -2), Q(4, 4), and R(-4, 5). Use analytic geometry to determine the coordinates of the orthocentre (the point where the altitudes intersect).

\triangle XYZ has vertices at X(0, 1), Y(6, -1), and Z(3, 6). Use analytic geometry to determine the coordinates of the centroid (the point where the medians intersect).

A new lookout tower is going to be built so that it is the same distance from three ranger stations. If the stations are at A(-90, 28), B(0, -35), and C(125, 20) on a grid, determine the coordinates of the point where the new tower should be built.

Predict the type of quadrilateral that is formed by the points of intersection of the lines 3x+y-4=0, 4x-5y+30=0, 4y = -3x -1, and -4x + 5y +10 = 0. Give reasons for your prediction. Verify that your prediction is correct by solving this problem.

A builder wants to run a temporary line from the main power line to a point near his site office. On the site plan, the site office is at S(25, 18) and the main power line goes through points T(1, 5) and U(29, 12). Each unit represents 1 m.

a. At what point should the builder connect to the main power line?

b. What length of cable will the builder need?