Secondary 4 Geometry Worksheet

16 of the 17 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 16 of the 17 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Angles of Depression and Elevation

   A delivery drone hovers motionless $48$ m above a level field. Its camera is aimed at a landing marker on the ground at an angle of depression of ${32}^{\circ }$. How far is the marker from the point on the ground directly beneath the drone? Round to the nearest tenth of a metre.

2. Q2Equivalent Figures

   A trapezoidal window pane has parallel sides of $9$ cm and $15$ cm and a height of $6$ cm. A rectangular pane $8$ cm wide is to be cut so that it is equivalent to the trapezoid. What must the rectangle's length be?

3. Q3Equivalent Solids

   A right circular cylinder has a radius of $5$ cm and a height of $12$ cm. A cone with the same radius is to be equivalent to this cylinder. Find the cone's height.

4. Q4Finding Missing Measurements in Plane Figures Using a 2nd Degree Equation

   A rectangular herb bed is $5$ m longer than it is wide and covers $84{\text{m}}^2$. Find its dimensions.

5. Q5Finding Missing Measurements in Plane Figures by Solving a System of Equations

   A rectangular banner has a perimeter of $48$ dm, and its length is $3$ dm more than twice its width. Find its dimensions.

6. Q6Geometry

   Most geometry problems are solved by first recognizing which relation applies. For each situation below, name the property or theorem you would use, and state the one condition that must be verified before you are allowed to use it.

   1. Two transversals cut three lines, and three of the four consecutive segment lengths they determine are known.
   2. An angle whose vertex is on a circle intercepts an arc whose measure is known.
   3. A cable runs from the top of a mast to an anchor on level ground, and the angle it makes with the ground is known.
   4. A container of one shape is to be replaced by a container of a different shape holding exactly the same amount.
7. Q7Methods to Decompose Figures

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

8. Q8Missing Measurements in Solids

   A right circular cone involves three lengths: the radius $r$ of the base, the height $h$, and the apothem (slant height) $s$.

   1. State the relation linking $r$, $h$ and $s$, and justify it.
   2. A technician knows only the volume and the radius of such a cone and must find its total area. Describe the order of the steps, naming which quantity must be found first and why the total area cannot be obtained directly.
9. Q9Similarity, Congruence, and Equivalence

   Decide whether each statement is always, sometimes or never true. Justify each answer with an argument or an explicit counterexample.

   1. Two congruent figures are equivalent.
   2. Two equivalent figures are congruent.
   3. Two similar figures whose ratio of similarity is $k=1$ are congruent.
   4. Two equivalent triangles have the same perimeter.
10. Q10Solids with the Same Area

    A cube has an edge of $6$ cm. A right prism with a square base $4$ cm by $4$ cm has the same total area as the cube. Find the height of the prism.

11. Q11Tetrahedrons

    A tetrahedron has an equilateral triangular base of side $8$ cm and a height of $15$ cm. Find its volume, exactly and to the nearest hundredth of a cubic centimetre.

12. Q12Thales' Theorem

    Three parallel lines are cut by two transversals. On the first transversal the two consecutive segments measure $6$ cm and $9$ cm. On the second transversal, the segment corresponding to the $6$ cm one measures $8$ cm. Find the length of the remaining segment.

13. Q13The Angles of a Circle

    In a circle of centre $O$, the chord $AB$ subtends a central angle $AOB$ of ${104}^{\circ }$.

    1. Point $C$ lies on the major arc $AB$. Find $m\angle ACB$.
    2. Find $m\angle OAB$.
14. Q14The Minimum Conditions for Congruent Triangles

    In triangles $ABC$ and $DEF$ it is known that $AB=DE=7$ cm, $BC=EF=9$ cm and $m\angle B=m\angle E={54}^{\circ }$.

    1. Name the minimum condition that establishes the congruence, and explain why the position of the angle matters.
    2. State the measure of $DF$ if $AC=8.2$ cm.
15. Q15The Minimum Conditions for Similar Triangles

    In triangles $PQR$ and $STU$, ${\displaystyle \frac{PQ}{ST}}={\displaystyle \frac{QR}{TU}}={\displaystyle \frac{3}{2}}$ and $m\angle Q=m\angle T$.

    1. Name the minimum condition that establishes the similarity.
    2. Find $PR$ if $SU=14$ cm.
16. Q16The Volume of Solids Using Algebra

    A rectangular-based right prism has dimensions $x$, $x+2$ and $x+5$ centimetres.

    1. Express its volume as a polynomial in expanded form.
    2. Evaluate the volume for $x=3$ cm and check the value directly.
17. Q17Synthesis — drawing on several sheets in this topic

    Road salt is dumped into a conical pile on level ground; the base of the pile is a circle of diameter $14$ m. From a point on the ground $12$ m from the edge of the pile, in the same vertical plane as the apex, the angle of elevation of the apex is ${22}^{\circ }$.

    The municipality wants to move all of the salt into a rectangular-based bin measuring $12$ m by $8$ m, filled level to the top. How deep must the bin be? Round to the nearest hundredth of a metre.

The 14 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.

Naming the relation is most of the work

A geometry question rarely tells you which theorem it wants. Q6 asks for exactly that, and adds a second demand that is where the marks quietly sit: the one condition that must be verified before you are allowed to use the relation. Every theorem in this unit has such a condition, and applying a theorem whose hypothesis fails is the most expensive kind of wrong answer, because the arithmetic afterwards is flawless and worth nothing.

Congruent, similar, equivalent — three different words (Q2, Q9, Q10)

They get used interchangeably in conversation and they mean three different things, which is why Q9 asks you to sort always, sometimes and never statements about them. Congruent means same shape and same size — every corresponding side and angle equal. Similar means same shape, sizes in a fixed ratio k. Equivalent means same area (or same volume for solids) and says nothing at all about shape.

The implications run one way only. Congruent figures are always similar and always equivalent. Similar figures with k = 1 are congruent. But equivalent figures need not be congruent, need not be similar, and need not even have the same number of sides — which is why Q2 can ask for a rectangle equivalent to a trapezoid at all.

One ratio worth carrying into every similarity question: if lengths are in the ratio k, then areas are in the ratio k² and volumes in the ratio k³. Doubling every edge of a solid multiplies its surface by 4 and its capacity by 8 — the reason this appears on exams as a "surprising" result is that most people expect all three to double.

Equivalent solids: change one dimension, not all of them (Q3)

Q3 sets a cone equal in volume to a cylinder of the same radius. Because a cone is exactly one third of the cylinder on the same base and height, matching their volumes with the radius held fixed forces the cone's height to be three times the cylinder's — a result you can predict before writing anything, and then confirm algebraically.

That prediction is the habit to build. Write the two volume formulas, cancel everything the two solids share (here π and r²), and see what is left. When only one dimension is free, the relation between it and the volume is linear and you solve directly. When the shape is fixed and all dimensions scale together — a sphere recast as a cone whose height is tied to its radius, for instance — the unknown ends up cubed, and a cube root appears in the answer. Knowing which of the two situations you are in tells you what the answer should look like before you get there.

Angles of elevation and depression (Q1, Q17)

Both angles are measured from the horizontal, never from the vertical: the angle of elevation looks up from the observer, the angle of depression looks down. The fact that makes the drone question work is that these two are equal — the horizontal at the drone and the ground are parallel, so the angle of depression at the top and the angle of elevation at the bottom are alternate interior angles. Draw that horizontal dashed line into your sketch and the right triangle appears on its own.

Thales, and the proportion that has to be built correctly (Q12)

Parallel lines cut proportional segments on any two transversals. The theorem is easy; setting up the proportion is where it goes wrong, because a proportion is only true if the two sides describe the same pair of positions. Write it as

with the left-hand fraction taken entirely from one transversal and the right-hand fraction entirely from the other, in the same order. Mixing a length from one transversal into the same fraction as a length from the other produces a plausible-looking equation with no theorem behind it. And read the question for what each length measures: a segment from the vertex is not the same as the whole side, and the two ratios AD⁄DB and AD⁄AB are different numbers.

The angles of a circle (Q13)

Three facts cover Q13 and most of what follows it. A central angle has the same measure as its intercepted arc. An inscribed angle, with its vertex on the circle, measures half its intercepted arc — so two inscribed angles standing on the same arc are equal, however far apart their vertices are. And any triangle formed by two radii is isosceles, because those two sides are both radii; that is what turns Q13(b) into a base-angle computation rather than a circle question at all.

Minimum conditions: what "minimum" is protecting you from (Q14, Q15)

A minimum condition is the smallest amount of information that pins a triangle down completely. For congruence: SSS, SAS, ASA. For similarity: AA, three proportional sides, or two proportional sides with the angle between them.

The word included in SAS is doing real work, and Q14 asks you to explain why. Two sides and the angle between them can be assembled in exactly one way — draw the angle, mark the two lengths along its arms, join the ends, and the triangle is determined. Move the given angle so that it is not between them and that guarantee vanishes: the third vertex can often be placed in two different positions, so two non-congruent triangles fit the same data. Once congruence is established, every corresponding side and angle is equal, which is all Q14(b) needs; once similarity is established, every pair of corresponding sides shares the same ratio k, which is all Q15(b) needs.

Building an equation for a missing measurement (Q4, Q5)

Both questions are algebra dressed as geometry, and both are won or lost in the first two lines. Name the unknown explicitly — "let w be the width in metres" — express every other length in terms of it, then write the one sentence in the problem that becomes an equation. Q4 turns into a second-degree equation because an area is a product of two lengths; Q5 gives two independent facts about two unknowns, so it becomes a system, most easily solved by substituting the second equation into the first.

Decomposing a figure, two ways on purpose (Q7)

The L-shaped floor is asked for twice — once by adding two rectangles, once by subtracting a rectangle from the enclosing one — and the reason is that the two methods check each other. If they disagree, one of your missing side lengths is wrong, and on an L-shape the missing sides are always found by subtracting: the short side of the notch is the difference of the two full sides it sits between. (This question depends on its diagram, so it lives on the printed sheet rather than on this page.)

The same discipline applies to any composite region. Decide whether the shape is a sum or a difference, mark the pieces on the drawing before computing anything, and treat a negative area as an immediate signal that you subtracted a piece larger than the whole.

Solids: find the bridge quantity first (Q8, Q11, Q16)

Q8 is the general principle stated in the open. In a right circular cone the radius, the height and the apothem are tied by s² = r² + h², because the axis, a radius and the corresponding slant line form a right triangle with the right angle at the centre of the base. Volume formulas contain h; area formulas contain s. So a question that gives you the volume and asks for the area has a compulsory middle step: get h from the volume, get s from Pythagoras, then compute the area. Naming that missing bridge quantity before you start is a strategy that generalises to every solid on this sheet.

Q11 needs the area of an equilateral triangle, (√3⁄4)a², before the pyramid volume ⅓ A_b h can be used — and the question asks for the exact value as well as the rounded one, so keep √3 intact through the whole computation and round only at the very end. Rounding early and then multiplying is how an answer drifts in the second decimal place. Q16 goes the other way, expressing a volume as a polynomial: multiply two brackets first, then distribute the third factor, and keep the units as cubic centimetres in the final line.

Q17: the synthesis problem

A conical salt pile measured by an angle of elevation, then moved into a rectangular bin. Three steps chained together — trigonometry for the height, the cone formula for the volume, equivalence for the depth of the bin — with one trap built into the wording. The angle is taken from a point measured from the edge of the pile, while the apex stands above its centre, so the horizontal distance in the right triangle is that measurement plus the radius. Sketch the vertical cross-section and the extra length is impossible to miss; skip the sketch and it is nearly impossible to catch.

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Getting the most out of it

Draw it, label it, then reach for a formula

Every question here that people get wrong, they get wrong at the diagram. Redraw the figure large, mark every given length and angle on it, and mark the unknown with a letter. The right relation usually becomes obvious once the picture is honest.

Answer the theorem-naming question in writing

The question that asks you to name a relation and state its condition, and the always/sometimes/never one, are the two most tempting to answer in your head. Write the sentences out. An exam wants the justification, not the verdict, and these are the questions that show whether you can produce one.

Keep exact values until the last line

Radicals and π survive the whole computation; rounding belongs to the final answer only. Several questions here ask for the exact form and the rounded form, which is a deliberate habit-building exercise for the exam.

Common questions