Given points A(1, 5), B(-3, 7) and C(15, 9).
(i) Find the equation of a line passing through the mid-point of AC and the point B.
(ii) Find the equation of the line through C and parallel to AB.
(iii) The lines obtained in part (i) and (ii) above, intersect each other at a point P. Find the coordinates of the point P.
(iv) Assign, giving reason, a special name of the figure PABC.

Question

Given points A(1, 5), B(-3, 7) and C(15, 9).
(i) Find the equation of a line passing through the mid-point of AC and the point B.
(ii) Find the equation of the line through C and parallel to AB.
(iii) The lines obtained in part (i) and (ii) above, intersect each other at a point P. Find the coordinates of the point P.
(iv) Assign, giving reason, a special name of the figure PABC.

KnowledgeBoat · Accepted Answer

(i) By formula, Mid-point (M) = Let D be the mid-point of AC. D = = (8, 7). By two-point formula, Equation of straight line : ⇒ y - y₁ = (x - x₁) Equation of BD : Hence, equation of a line passing through the mid-point of AC and the point B is y - 7 = 0. (ii) By formula, Slope = Slope of AB = . As, slope of parallel lines are equal. ∴ Slope of line parallel to AB = -. By point-slope form, equation : Slope of line through C and parallel to AB. Hence, equation of the line through C and parallel to AB is x + 2y = 33. (iii) Equations are : ⇒ y - 7 = 0 ..........(1) ⇒ x + 2y = 33 .........(2) From equation (1), ⇒ y = 7 ........(3) Substituting above value of y in equation (2), we get : ⇒ x + 2(7) = 33 ⇒ x + 14 = 33 ⇒ x = 33 - 14 = 19. ∴ P = (x, y) = (19, 7). Hence, P = (19, 7). (iv) Steps : 1. Mark points A, B, C and P. 2. Join AB, BC, CP and PA. 3. From graph, PABC is a trapezium. Hence, PABC is a trapezium.

Mathematics

Straight Line Eq

Related Questions

Show that the points (a, b), (a + 3, b + 4), (a - 1, b + 7) and (a - 4, b + 3) are the vertices of a parallelogram.

A(-4, 2), B(0, 2) and C(-2, -4) are vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB, respectively. Show that the centroid of △PQR is the same as the centroid of △ABC.

The line x - 4y = 6 is the perpendicular bisector of the line segment AB. If B = (1, 3); find the coordinates of point A.

Find the equation of a line passing through the points (7, -3) and (2, -2). If this line meets x-axis at point P and y-axis at point Q; find the co-ordinates of points P and Q.