Answer Key
9.5 Coordinate Geometry
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | State the gradient $m$ and the $y$-intercept $c$ of the line $y = 3x + 2$. | $m = 3, c = 2$ |
| 2 | Find the gradient of the line through $(1, 2)$ and $(4, 8)$. | 2 |
| 3 | Find the midpoint of the segment from $(2, 4)$ to $(8, 10)$. | $(5, 7)$ |
| 4 | Find the distance between $(0, 0)$ and $(3, 4)$. | 5 |
| 5 | A line parallel to $y = 4x + 7$. Find its gradient. | 4 |
| 6 | A line perpendicular to $y = 2x + 3$. Find its gradient. | $-1/2$ |
| 7 | Complete the table of values for $y = 2x + 1$ at $x = -1, 0, 1, 2$. | $-1, 1, 3, 5$ |
| 8 | Write the equation of the line with gradient 3 and $y$-intercept -4. | $y = 3x - 4$ |
| 9 | State the equation of the horizontal line through $(2, 7)$ and the vertical line through $(-3, 5)$. | Horizontal: $y = 7$; vertical: $x = -3$ |
| 10 | Find the $x$- and $y$-intercepts of $y = 3x + 6$. | $x = -2$, $y = 6$ |
| 11 | Find the gradient through $(3, 8)$ and $(7, 2)$. | $-3/2$ |
| 12 | Find the distance between $(1, 1)$ and $(4, 5)$ as a simplified surd. | 5 |
| 13 | Find the equation of the line through $(1, 3)$ and $(3, 7)$. | $y = 2x + 1$ |
| 14 | Find the equation of the line parallel to $y = 2x + 3$ passing through $(1, 4)$. | $y = 2x + 2$ |
| 15 | Find the equation of the line perpendicular to $y = 2x + 1$ through $(4, 3)$. | $y = -x/2 + 5$ |
| 16 | Find the midpoint of $A(-4, 3)$ and $B(6, -7)$ with mixed signs. | $(1, -2)$ |
| 17 | Find the equation of the line with gradient 3 through $(2, 1)$. | $y = 3x - 5$ |
| 18 | Decide parallel / perpendicular / neither: (a) $y = 3x + 2$ and $y = 3x - 5$. (b) $y = 4x + 1$ and $y = -x/4 + 5$. | (a) parallel (b) perpendicular |
| 19 | Find the equation of the line perpendicular to $y = (1/3)x + 2$ passing through $(3, 1)$. | $y = -3x + 10$ |
| 20 | Find the equation of the line through (2, 4) and (5, 4). | $y = 4$ (horizontal) |
| 21 | A line passes through $A(-2, 1)$ and $B(7, 4)$. Find its equation. | $y = (1/3)x + 5/3$ |
| 22 | The point C divides AB in ratio 2:1, $A(-2, 1)$, $B(7, 4)$. Find C. | $C = (4, 3)$ |
| 23 | Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6). | $y = -x + 7$ |
| 24 | Find the $x$- and $y$-intercepts of $3x + 4y = 12$. | x-int (4, 0); y-int (0, 3) |
| 25 | Does the point (3, 7) lie on the line $y = 2x + 1$? Justify. | Yes ($7 = 7$) |
| 26 | Find the equation of the line parallel to $2x + 3y = 6$ passing through $(0, 0)$. | $y = -(2/3)x$ |
| 27 | Find the perimeter of triangle $A(0, 0)$, $B(4, 3)$, $C(8, 0)$. | $5 + 5 + 8 = 18$ |
| 28 | A line $\ell_1$ has equation $y = mx + 2$. It is perpendicular to $\ell_2: y = (1/4)x + 5$. Find $m$. | $m = -4$ |
| 29 | Find the foot of the perpendicular from $P(5, 1)$ to the line $y = 2x - 1$. | $F = (9/5, 13/5)$ |
| 30 | A line through $(1, 5)$ is parallel to a line through $(0, 1)$ and $(4, 9)$. Find its equation. | $y = 2x + 3$ |
| 31 | A triangle has vertices A(1, 2), B(7, 4), C(4, 10). (a) Find the midpoint M of BC. (b) Find the equation of the median from A. | (a) (5.5, 7) (b) $y = (10/9)x + 8/9$ |
| 32 | Show that points P(-1, -3), Q(2, 3), R(5, 9) are collinear. | Yes — gradients PQ = QR = 2 |
| 33 | Find the perpendicular distance from $P(5, 1)$ to the line $y = 2x - 1$. | $\dfrac{8\sqrt{5}}{5}$ |
| 34 | A line passes through $(a, 0)$ and $(0, b)$. Find its equation. | $y = -(b/a)x + b$ (or $x/a + y/b = 1$) |
| 35 | Find the area of the triangle bounded by $y = x + 1$, $y = -x + 5$, $y = 0$. | 9 sq units |
| 36 | A triangle has vertices $A(0, 0), B(4, 2), C(1, 5)$. Verify whether it is right-angled. | Not right-angled (no perpendicular pairs) |
| 37 | The point M(3, 2) is the midpoint of AB, where A = (1, 5). Find B. | $B = (5, -1)$ |
| 38 | A quadrilateral has vertices A(0, 0), B(4, 0), C(5, 3), D(1, 3). Find the gradients of all 4 sides and identify the shape. | AB 0, BC 3, CD 0, DA 3. Parallelogram (parallel sides equal). |
| 39 | Find the value of $k$ for which the lines $y = 2x + k$ and $y = 3x + 1$ intersect at a point on the $x$-axis. | $k = -2/3$ (after computation) |
| 40 | The line $\ell$ has equation $3x - 4y = 12$. Find (a) the slope, (b) the $x$- and $y$-intercepts, (c) the perpendicular distance from origin to $\ell$. | (a) $3/4$ (b) $x$-int (4, 0); $y$-int (0, -3) (c) $12/5$ |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | State the gradient $m$ and the $y$-intercept $c$ of the line $y = -2x + 5$. | $m = -2, c = 5$ |
| 2 | Find the gradient of the line through $(0, 3)$ and $(5, 13)$. | 2 |
| 3 | Find the midpoint of the segment from $(-1, 3)$ to $(5, 7)$. | $(2, 5)$ |
| 4 | Find the distance between $(1, 2)$ and $(7, 10)$. | 10 |
| 5 | A line parallel to $y = -3x + 1$. Find its gradient. | $-3$ |
| 6 | A line perpendicular to $y = 3x + -1$. Find its gradient. | $-1/3$ |
| 7 | Complete the table of values for $y = -1x + 4$ at $x = -1, 0, 1, 2$. | $5, 4, 3, 2$ |
| 8 | Write the equation of the line with gradient -2 and $y$-intercept 5. | $y = -2x + 5$ |
| 9 | State the equation of the horizontal line through $(2, -2)$ and the vertical line through $(4, 5)$. | Horizontal: $y = -2$; vertical: $x = 4$ |
| 10 | Find the $x$- and $y$-intercepts of $y = 3x + 6$. | $x = 4$, $y = 8$ |
| 11 | Find the gradient through $(-2, 5)$ and $(4, -7)$. | $-2$ |
| 12 | Find the distance between $(1, 1)$ and $(4, 5)$ as a simplified surd. | $5\sqrt{2}$ |
| 13 | Find the equation of the line through $(0, -1)$ and $(4, 11)$. | $y = 3x - 1$ |
| 14 | Find the equation of the line parallel to $y = 3x + -1$ passing through $(2, 5)$. | $y = 3x - 1$ |
| 15 | Find the equation of the line perpendicular to $y = 3x + -2$ through $(6, 1)$. | $y = -x/3 + 3$ |
| 16 | Find the midpoint of $A(-5, -2)$ and $B(9, 8)$ with mixed signs. | $(2, 3)$ |
| 17 | Find the equation of the line with gradient -2 through $(-1, 4)$. | $y = -2x + 2$ |
| 18 | Decide parallel / perpendicular / neither: (a) $y = 3x + 2$ and $y = 3x - 5$. (b) $y = 4x + 1$ and $y = -x/4 + 5$. | Same. |
| 19 | Find the equation of the line perpendicular to $y = (1/3)x + 2$ passing through $(3, 1)$. | $y = -2x + 8$ |
| 20 | Find the equation of the line through (2, 4) and (5, 4). | $x = 3$ (vertical) |
| 21 | A line passes through $A(-2, 1)$ and $B(7, 4)$. Find its equation. | $y = -(1/3)x + 1$ |
| 22 | The point C divides AB in ratio 2:1, $A(-2, 1)$, $B(7, 4)$. Find C. | $C = (1, 4)$ |
| 23 | Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6). | $y = (3/2)x - 1/2$ |
| 24 | Find the $x$- and $y$-intercepts of $3x + 4y = 12$. | x-int (4, 0); y-int (0, 10) |
| 25 | Does the point (3, 7) lie on the line $y = 2x + 1$? Justify. | Yes ($5 = 5$) |
| 26 | Find the equation of the line parallel to $2x + 3y = 6$ passing through $(0, 0)$. | $y = (3/5)x + 2/5$ |
| 27 | Find the perimeter of triangle $A(0, 0)$, $B(4, 3)$, $C(8, 0)$. | $10 + 10 + 12 = 32$ |
| 28 | A line $\ell_1$ has equation $y = mx + 2$. It is perpendicular to $\ell_2: y = (1/4)x + 5$. Find $m$. | $m = 3$ |
| 29 | Find the foot of the perpendicular from $P(5, 1)$ to the line $y = 2x - 1$. | Same. |
| 30 | A line through $(1, 5)$ is parallel to a line through $(0, 1)$ and $(4, 9)$. Find its equation. | $y = 2x - 5$ |
| 31 | A triangle has vertices A(1, 2), B(7, 4), C(4, 10). (a) Find the midpoint M of BC. (b) Find the equation of the median from A. | (a) (4, 4) (b) $y = (1/5)x + 16/5$ |
| 32 | Show that points P(-1, -3), Q(2, 3), R(5, 9) are collinear. | Same. |
| 33 | Find the perpendicular distance from $P(5, 1)$ to the line $y = 2x - 1$. | 0 |
| 34 | A line passes through $(a, 0)$ and $(0, b)$. Find its equation. | Same. |
| 35 | Find the area of the triangle bounded by $y = x + 1$, $y = -x + 5$, $y = 0$. | Same. |
| 36 | A triangle has vertices $A(0, 0), B(4, 2), C(1, 5)$. Verify whether it is right-angled. | Same. |
| 37 | The point M(3, 2) is the midpoint of AB, where A = (1, 5). Find B. | $B = (5, 7)$ |
| 38 | A quadrilateral has vertices A(0, 0), B(4, 0), C(5, 3), D(1, 3). Find the gradients of all 4 sides and identify the shape. | Same. |
| 39 | Find the value of $k$ for which the lines $y = 2x + k$ and $y = 3x + 1$ intersect at a point on the $x$-axis. | Pack B answer: $k = -8/3$ |
| 40 | The line $\ell$ has equation $3x - 4y = 12$. Find (a) the slope, (b) the $x$- and $y$-intercepts, (c) the perpendicular distance from origin to $\ell$. | (a) $-5/12$ (b) $x$-int (12, 0); $y$-int (0, 5) (c) $60/13$ |
Problems — Worked Solutions
**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.
(a) $\tfrac{1}{3}$ (b) $y = \tfrac{1}{3}x + \tfrac{5}{3}$ (c) $y = \tfrac{1}{3}x$
**Dividing a segment.** Point C divides AB in ratio 2:1, where $A(-2, 1)$, $B(7, 4)$. Find C.
$C = (4, 3)$
**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.
(a) 10 km (b) (6, 8) (c) $y = -(4/3)x + 16$
**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.
(a) P(3, 0), Q(7, 2), R(5, 5), S(1, 3) (b) PQ ∥ SR (gradient 1/2); QR ∥ PS (gradient $-3/2$)
**Perpendicular bisector.** Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).
$y = -x + 7$
**Triangle on coordinate plane.** Find the perimeter of the triangle with vertices A(0, 0), B(4, 3), C(8, 0).
18
**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.
(a) $y = -x/2 + 7/2$ (b) $F = (9/5, 13/5)$ (c) $\tfrac{8\sqrt{5}}{5}$
**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?
(a) $m = 3/2$ (b) $m = (3 \pm \sqrt{13})/2$
**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.
(a) $(-1, 0)$, $(5, 0)$, $(2, 3)$ (b) 9
**Identifying a parallelogram.** Show that the points A(1, 1), B(4, 1), C(6, 4), D(3, 4) form a parallelogram.
AB ∥ DC (both horizontal). AD ∥ BC (gradient 3/2)
**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.
(a) (3, 2) (b) See working
**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.
(a) $y = -2x - 1$ (b) $y = -2x + 1$