9.5 Coordinate Geometry

Montrez tous les calculs. Des points partiels sont accordés pour la méthode.

1

**Line through two points.** The line AB passes through $A(-2, 1)$ and $B(7, 4)$. (a) Find the gradient as a fraction in simplest form. (b) Find the equation in $y = mx + c$ form. (c) State the equation of a line parallel to AB through the origin.

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2

**Dividing a segment.** Point C divides AB in ratio 2:1, where $A(-2, 1)$, $B(7, 4)$. Find C.

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3

**Distance and midpoint together.** Two villages at $A(2, 5)$ and $B(10, 11)$ on a map (km). (a) Find the straight-line distance. (b) Find the midpoint. (c) A new road perpendicular to AB through the midpoint. Find its equation.

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4

**Mid-segment investigation.** A quadrilateral with vertices A(0, 0), B(6, 0), C(8, 4), D(2, 6). (a) Find the midpoints P, Q, R, S of sides AB, BC, CD, DA. (b) Show PQRS is a parallelogram by comparing gradients.

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5

**Perpendicular bisector.** Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).

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6

**Triangle on coordinate plane.** Find the perimeter of the triangle with vertices A(0, 0), B(4, 3), C(8, 0).

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7

**Distance and perpendicular distance.** A point $P(5, 1)$ and a line $y = 2x - 1$. (a) Find the equation of the line through P perpendicular to the given line. (b) Find the foot of the perpendicular (i.e. the point on the line closest to P). (c) Find the perpendicular distance from P to the line.

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8

**Investigating parallel/perpendicular.** Lines $\ell_1: y = mx + 2$ and $\ell_2: y = (3 - m)x + 5$. (a) For what value of $m$ are $\ell_1$ and $\ell_2$ parallel? (b) For what value(s) of $m$ are they perpendicular?

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9

**Triangle bounded by three lines.** Sketch the triangle bounded by $y = x + 1$, $y = -x + 5$, $y = 0$. (a) Find the three vertices. (b) Find the area.

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10

**Identifying a parallelogram.** Show that the points A(1, 1), B(4, 1), C(6, 4), D(3, 4) form a parallelogram.

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11

**Centroid of a triangle.** A triangle has vertices A(0, 0), B(6, 0), C(3, 6). (a) Find the centroid (average of vertices). (b) Find the medians and verify they all pass through the centroid.

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12

**Reflection of a line.** The line $y = 2x + 1$ is reflected in the $x$-axis. (a) Find the equation of the image. (b) Find the equation when reflected in the $y$-axis.

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