Secondary 4 Analytic Geometry Worksheet

All 14 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Analytic Geometry

   Analytic geometry answers geometric questions with coordinates and algebra. For each situation below, name the analytic tool you would use (a formula or a condition on slopes), and say what the tool produces — a number, a pair of coordinates, or a yes/no verdict.

   1. Deciding whether two segments of a roof truss meet at a right angle.
   2. Locating the point three-fifths of the way along a segment from $A$ to $B$.
   3. Showing that a triangle drawn on a plan is isosceles.
   4. Deciding whether a house at $(4,-1)$ lies on the protected side of a straight shoreline.
2. Q2Forms of the Equation of a Line

   A conveyor belt is modelled by the line $3x+4y-24=0$.

   1. Rewrite it in functional form and state the slope.
   2. Rewrite it in symmetric form and state both intercepts.
3. Q3Proofs in Analytic Geometry

   To prove that the diagonals of a rhombus are perpendicular, a student draws the rhombus with vertices $(0,0)$, $(5,0)$, $(8,4)$ and $(3,4)$, checks that all four sides measure $5$ units, computes the slopes of the two diagonals as $\frac{1}{2}$ and $-2$, notes that the product is $-1$, and writes “proved”. Explain why this is not a proof, and describe precisely how the coordinates should be chosen so that the same computation does establish the general statement.

4. Q4Straight Lines and Half-Planes

   The line $2x-3y=6$ cuts the plane into two half-planes. For each set below, say whether it is a line, an open half-plane, a closed half-plane, or none of these, and justify in one sentence.

   1. $\{(x,y)\mid 2x-3y>6\}$
   2. $\{(x,y)\mid 2x-3y=6\}$
   3. $\{(x,y)\mid |2x-3y|\ge 6\}$
   4. $\{(x,y)\mid y\le 4\}$
5. Q5The Distance Between 2 Points

   On a climbing wall, coordinates are given in metres. An anchor bolt sits at $A(-3,7)$ and another at $B(9,2)$. How long a rope is needed to run straight from one bolt to the other?

6. Q6The Distance Between Two Parallel Lines

   Find the distance between the parallel lines $y=\frac{3}{4}x+2$ and $y=\frac{3}{4}x-3$.

7. Q7The Distance From a Point to a Line on a Cartesian Plane

   Find the distance from the point $P(7,-2)$ to the line $4x-3y+1=0$.

8. Q8The Division Point and Midpoint of a Segment

   A straight ski trail runs from the chalet $A(-6,2)$ to the summit $B(6,10)$ (units in hundreds of metres).

   1. Find the midpoint $M$ of $\overline{AB}$.
   2. A rest bench is placed one quarter of the way from $A$ to $B$. Find its coordinates.
9. Q9The Equation of a Line From the Coordinates or Slope

   Find the equation, in functional form, of the line passing through $(-2,11)$ and $(4,-7)$.

10. Q10The Graph of a Line

    Draw the line $2x+3y=12$ on the grid below by first plotting its two intercepts, then state its slope.

    A blank Cartesian grid for this question is on the printable PDF.

11. Q11The Half-Plane and the Solution Set

    Consider the inequality $2x-y\le 4$.

    1. Give the equation of the boundary line and say whether it is drawn solid or dashed.
    2. Use a test point to decide which side is shaded, then sketch the solution set on the grid.

    A blank Cartesian grid for this question is on the printable PDF.

12. Q12The Relative Position of Two Lines

    Three lines are given: ${\ell }_1:3x-6y+5=0$, ${\ell }_2:y=\frac{1}{2}x-4$ and ${\ell }_3:2x+y=7$. Determine the relative position of each of the three pairs $({\ell }_1,{\ell }_2)$, $({\ell }_1,{\ell }_3)$ and $({\ell }_2,{\ell }_3)$.

13. Q13The Slope of a Line

    A wheelchair ramp rises $0.6$ m over a horizontal run of $7.2$ m.

    1. Give its slope as a fraction in lowest terms.
    2. Express that slope as a percent, to the nearest hundredth.
14. Q14Synthesis — drawing on several sheets in this topic

    On a site plan graduated in metres, a straight drainage pipe runs from $A(-4,-2)$ to $B(8,4)$. A maintenance hatch $H$ is installed on the pipe, two thirds of the way from $A$ to $B$. A protected oak stands at $T(0,9)$.

    1. Find the coordinates of $H$ and the total length of the pipe, to the nearest hundredth of a metre.
    2. By-law: no tree may be closer than $6$ m to a pipe. Does the oak comply? Support your answer with a computation.

The 11 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Choosing the tool is the actual skill

Analytic geometry is a small box of formulas and one large decision: which one this question wants. Q1 asks that decision directly, with no numbers at all, and it is the best predictor on the sheet of how the exam will go. Work from what the question wants out, not from what it gives you.

The three forms of a line, and why anyone needs three (Q2)

They are the same line written for three different purposes, and switching between them is a mechanical skill worth practising until it is boring.

General form Ax + By + C = 0 puts everything on one side. It is the form every distance formula demands, and the only one that can describe a vertical line. Functional form y = ax + b isolates y and hands you the slope and the y-intercept by inspection. Symmetric form x⁄a + y⁄b = 1 hands you both intercepts by inspection, which is why it is the form to reach for when a question is about where a line meets the axes.

One relation is worth memorising rather than re-deriving each time: in general form the slope is −A⁄B. It falls straight out of isolating y, and it saves a rearrangement every time a question hands you a line as Ax + By + C = 0 and asks about direction — which Q12 does three times over.

The two distance formulas people mix up (Q5, Q6, Q7)

Between two points you use coordinates only. Between a point and a line you use the coefficients of the line and the coordinates of the point. They look nothing alike and they still get swapped, usually because the line was left in functional form.

Q6 asks for the distance between two parallel lines, and there is no separate formula for it. Pick any point you like on one line — the easiest is usually its y-intercept — and measure from that point to the other line. Because the lines are parallel, the answer does not depend on which point you picked, and that is the whole reason the trick works. If the two lines you were handed do not share the same A and B after simplifying, they are not parallel and the question has a different answer.

Slope, and the four verdicts it gives (Q12, Q13)

Slope is rise over run, a = (y₂ − y₁)⁄(x₂ − x₁), and Q13 asks for it as a fraction in lowest terms and again as a percent — the two ways construction and engineering write the same number. A percent grade is simply the slope multiplied by 100, so a slope of one twelfth is a grade of about 8.33%.

For a pair of lines, compute both slopes and read the verdict. Equal slopes with different y-intercepts means parallel and distinct — no intersection. Equal slopes and equal intercepts means the same line drawn twice, and every point is a solution. A product of −1 means perpendicular. Anything else means they cross exactly once. The case to watch is a vertical line: it has no slope at all, so the product test cannot be applied, and a vertical and a horizontal line are perpendicular by inspection instead.

Division points: k is measured from A (Q8, Q14)

The formula P = A + k(B − A) reads as "start at A and walk the fraction k of the way toward B". Applied to each coordinate separately, that is all there is to it, and the midpoint is just the case k = ½.

Half-planes: the boundary and the side (Q4, Q11)

An inequality in two variables describes a region, and describing it takes two decisions. First, the boundary: draw it solid for ≤ or ≥, because those points belong to the set, and dashed for < or >, because they do not. That is the difference between a closed and an open half-plane, and Q4 asks you to name it.

Second, the side. Take any point not on the boundary, substitute it, and see whether the statement is true; if it is, shade that side. The origin is the easiest test point in existence — unless the boundary passes through it, in which case pick something else. Do not try to reason it out from the direction of the inequality sign; the sign flips meaning depending on which side of the equation y ended up, and the test point never lies.

Q4 also includes a set that is not a half-plane, and the reason is a definition worth stating precisely: a half-plane is bounded by one line. A condition built from an absolute value produces two boundary lines and the region outside a strip between them, so it fails the definition no matter how plausible it looks.

What makes a coordinate argument a proof (Q3)

Q3 is the hardest question on the sheet and the one most students hand in half-done, because checking a specific rhombus feels like work and reads like a proof. It is not one. Verifying a single case cannot establish a claim made about every figure; it only fails to contradict it.

Q14: the synthesis problem

A pipe from A to B, a hatch two thirds of the way along it, and a protected tree that must be far enough from the pipe. Three tools in one question: the division point for the hatch, the distance formula for the pipe's length, and the point-to-line distance for the clearance — and that last one needs the equation of the pipe, which you build from its two endpoints first. Read the by-law carefully before you conclude: a clearance comfortably larger than the minimum and one comfortably smaller both need the same sentence of interpretation to earn the mark.

Preview all 6 pages

Click any page to open the full PDF.

Getting the most out of it

Draw the axes even when the question looks purely algebraic

A rough sketch with the points plotted takes twenty seconds and catches almost every sign error on this sheet — a midpoint outside its segment, a distance that came out larger than the whole line, a half-plane shaded on the wrong side.

Convert to general form once, at the start

Both distance formulas and the slope shortcut all read from Ax + By + C = 0. Rewriting every line the question gives you before you begin removes the most common source of wrong coefficients.

Come back to the tool-naming question at the end

Q1 asks which tool applies and what it produces, without computing anything. Do it once before you start and once after you finish; the second attempt is a summary of the whole sheet, and exam questions are usually phrased that way rather than as bare formulas.

Common questions