Secondary 3 Missing Measurements, Projections and Lines

9 of the 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 9 of the 12 questions are printed below. The other 3 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

1. Q1Central Projections With One or Two Vanishing Points

   This question is built around a diagram or a table of values. Open it in the PDF.

2. Q2Finding Missing Measurements in Plane Figures by Solving a System of Equations

   A rectangular community garden in Rosemont is enclosed by 96 m of fencing. Its length is 8 m less than three times its width.

   1. Using $L$ for the length and $w$ for the width, write a system of two first-degree equations.
   2. Solve the system by substitution.
   3. Give the area of the garden, and check your two dimensions against both of the original statements.
3. Q3Missing Measurements According to Area: Decomposable and Truncated Solids

   A concrete bollard at the entrance of a parking lot is made of two stacked rectangular prisms: a lower block with a square base of side 12 cm and unknown height $h$, and, centred on top of it, a cube of edge 6 cm. The whole outside surface — including the base it stands on — is to be painted, and it takes $720{\text{cm}}^2$ of paint coverage.

   1. Write the total exterior area of the solid as an expression in $h$. Remember that the top of the lower block is only partly exposed.
   2. Find $h$.
4. Q4Missing Measurements in Solids

   Each situation below asks for one missing measurement of a solid. For each one, (i) name the formula you would turn into an equation, (ii) say how many unknowns that equation contains, and (iii) find the measurement or explain why it cannot be found from what is given.

   1. A cube has a total surface area of $294{\text{cm}}^2$. Find its edge.
   2. A cylinder has a volume of $500\pi {\text{cm}}^3$ and a height of 20 cm. Find its radius.
   3. A cone has a volume of $96\pi {\text{cm}}^3$. Find its radius.
   4. A rectangular prism with a square base has a volume of $240{\text{cm}}^3$ and a height of 15 cm. Find the side of its base.
5. Q5Missing Measurements of Solids from the Area

   Find the missing measurement in each solid. Work with $\pi$ as a symbol rather than as a decimal, and give exact answers.

   1. A closed cylindrical soup can has a radius of 5 cm and a total surface area of $130\pi {\text{cm}}^2$. Find its height.
   2. A cone-shaped party hat (no base) has a slant height of 12 cm and a lateral area of $96\pi {\text{cm}}^2$. Find the radius of its opening.
   3. A ball has a surface area of $144\pi {\text{cm}}^2$. Find its radius.
6. Q6Missing Measures from a Volume: Decomposable and Truncated Solids

   A garden shed is a decomposable solid: a rectangular prism 6 m long and 4 m wide, of unknown height $h$, with a roof in the shape of a triangular prism sitting on top. The roof's triangular face has a base of 4 m (the width of the shed) and a height of 1.5 m, and the roof is 6 m long, like the shed. The whole shed encloses $90{\text{m}}^3$ of air.

   1. Find the volume of the roof.
   2. Find $h$, the height of the walls.
   3. How tall is the shed from the ground to the ridge of the roof?
7. Q7Orthogonal Projections (Multiple Views)

   This question is built around a diagram or a table of values. Open it in the PDF.

8. Q8Parallel Projections (Cavalier and Axonometric)

   This question is built around a diagram or a table of values. Open it in the PDF.

9. Q9Projections and Perspectives

   Each part below describes a drawing; no drawing is printed here. Name the type of projection each one uses — orthogonal projection (multiple views), cavalier perspective, isometric (axonometric) perspective, central perspective with one vanishing point, or central perspective with two vanishing points — and give the clue that decides it.

   1. A plan of a garage made of three separate drawings labelled front, top and right side, each carrying true measurements.
   2. A drawing of a bookcase whose front face is a true-size rectangle and whose depth edges are all at ${45}^{\circ }$ and half their real length.
   3. A drawing of a metro corridor in which the floor lines, the ceiling lines and the tops of all the doors meet at a single point.
   4. A drawing of a cube in which the edges run along three axes at ${120}^{\circ }$ to one another and no face is a square.
   5. A drawing of a street corner in which the vertical edges stay vertical while the two visible walls recede toward two different points on the horizon.

   Finally: which of the five keep parallel edges of the solid parallel on the drawing?

10. Q10The Relative Position of Two Lines

    In the Cartesian plane, line $d_1$ passes through $A(-2,1)$ and $B(4,5)$.

    1. Find the slope of $d_1$.
    2. Line $d_2$ passes through $C(0,-3)$ and $D(9,3)$. Find its slope, then state the relative position of $d_1$ and $d_2$. Justify why they are not the same line.
    3. Line $d_3$ passes through $E(2,5)$ and $F(6,-1)$. Find its slope, then state the relative position of $d_1$ and $d_3$, with the reason.
    4. Line $d_4$ passes through $G(1,3)$ and $H(7,7)$. Find its slope, then state the relative position of $d_1$ and $d_4$. Be careful — equal slopes are not the whole story.
11. Q11The Slope of a Line

    Find the slope in each case. Give exact values, and say what the sign of each slope means.

    1. The line through $M(-4,7)$ and $N(2,-5)$.
    2. An access ramp that rises 24 cm over a horizontal run of 3 m. (Read the units carefully.)
    3. The line through $T(3,-2)$ and $U(7,-2)$.
    4. The line through $V(5,1)$ and $W(5,9)$.
12. Q12Synthesis — drawing on several sheets in this topic

    A skatepark is getting a new concrete launch ramp. Seen from the side it is a right triangle: the sloping surface rises 1 m over a horizontal run of 4 m. The ramp is a triangular prism, and its width $w$ (across the direction skaters travel) is still to be decided.

    1. Find the slope of the ramp's sloping surface, as a fraction and as a decimal.
    2. The city will pour exactly $6{\text{m}}^3$ of concrete. Find $w$.
    3. Describe the ramp's top view and its right side view, with dimensions.
    4. A safety rule says the ramp's slope must not exceed $0.3$. If the run stayed at 4 m, what is the greatest rise allowed? Would a rise of 1.5 m pass?

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.

Counting the unknowns is the whole method (Q4)

Q4 is the sheet's key question and it is deliberately not about arithmetic. For each situation you name the formula, say how many unknowns it still contains, and only then solve — or explain why you cannot. One equation settles one unknown. Two unknowns need a second, independent equation, and no amount of algebra will conjure it out of the first.

Q5 runs that procedure three times on areas, and each one comes out as a first-degree or a simple squared equation. Keep π as a symbol: it divides out cleanly and the answers stay exact.

Two unknowns, two equations (Q2)

This is the year the system of two first-degree equations arrives, and geometry is where it earns its keep. Q2's garden gives two facts about two dimensions — a perimeter, and a relation between length and width — so it becomes a system, most easily solved by substitution: the relation already expresses one variable in terms of the other, so put it into the perimeter equation and one unknown disappears.

Composite solids: build the expression, then solve it (Q3, Q6)

Q3 and Q6 are the same idea applied to area and to volume. A solid is made of two pieces, one dimension is unknown, and the total is given. Write the total as an expression in that unknown — piece by piece, on separate lines — set it equal to the given total, and solve a first-degree equation.

The care goes into the expression, not the algebra. In Q3 the top of the lower block is only partly exposed, because the cube sits on part of it, so that face contributes its area minus the covered square. In Q6, by contrast, nothing is hidden: volumes simply add, and the roof can be computed on its own and subtracted from the total before the walls are touched. Area questions hide surfaces; volume questions do not.

What each projection keeps, and what it throws away (Q1, Q7, Q8, Q9)

Four questions here are about drawings, and they all turn on one comparison. Every projection gives up something; which one you choose depends on what you cannot afford to lose.

Q1 pins down the one-point case exactly: the front face keeps its right angles, its parallel sides and its measurements, because it lies flat against the picture; the back face keeps the shape but not the size; the four depth edges each run along the line joining their corner to V. Q8 pins down the cavalier case: the eight edges of the front and back faces are true lengths and the four receding ones are not, so a drawn 1.5 cm at a reduction factor of ½ stands for 3 cm of real depth.

Orthogonal views show the silhouette, not the inside (Q7)

Q7's block has a step in it, yet its right side view is a plain rectangle. That is not an error in the drawing: a view is the outline of what you see from that direction, so the tall part and the low part overlap and together fill the whole rectangle. An internal line is added to mark the edge where they meet, but the outside boundary is the union of the two silhouettes.

The same fact has a consequence worth carrying: three views give the overall width, depth and height and the outline from each direction, but they can hide a cavity that reaches no face. That is why technical drawings add dashed hidden edges or a cross-section.

Slope: rise over run, and the two special cases (Q11)

The slope of a line is the rise divided by the run between any two of its points, and its sign carries meaning: positive rises to the right, negative falls. Q11 adds three things that are easy to get wrong.

Equal slopes are not enough (Q10)

Q10 works through four lines and makes one point four times over. Same slope means the lines point in the same direction — but two lines can point the same way because they are parallel, or because they are the same line. Part (d) is exactly that case, and the only way to tell is to test whether a point of one lies on the other.

One limitation is worth stating, because a later question depends on it: a slope describes a direction and knows nothing about length. Slopes can prove that sides are parallel or perpendicular, so they can establish a parallelogram or a right angle — but they can never show that two sides are equal. For that you need the lengths, and the Pythagorean relation, which arrives this year, is how you get them from horizontal and vertical displacements.

Q12: the synthesis question

A skatepark ramp, seen as a triangular prism. Its side profile gives a slope; its volume gives the missing width; its top and side views are two of the orthogonal projections from earlier on the sheet; and a safety limit on the slope turns into an inequality about the greatest rise allowed for a fixed run. Four strands of the sheet in one object, and the working is short if you take them in that order.

Getting the most out of it

Count the unknowns before you try to solve

Substitute everything you know into the formula, then look at what is left. One unknown means solve; two means find a second fact in the wording or say that the measurement cannot be found. Recognising the second case is a full answer, not a failure.

Name the variables in words

"Let w be the width in metres." Every system question on this sheet is easier after that line, and the check at the end — testing your pair against both original statements — is what catches a substitution slip while it is still cheap.

Ask what the drawing is for

Before naming a projection, ask whether the reader needs to measure it or to recognise it. Measuring needs a parallel projection with a stated scale; recognising needs a central perspective, which by construction cannot be measured. That single question answers most of the projection items on this sheet.

Common questions