Chapter 7 — Geometry and Trigonometry

Answer: A

Explanation

The three angles of a triangle add to $180°\mathclose{:}$

$$180 - 47 - 68 = 65$$

Check: $47 + 68 + 65 = 180$ ✓

Why each wrong option is wrong:

• B, $112$ — subtracted only the $68°$ from 180 and forgot the $47°\mathclose{.}$
• C, $115$ — added the two given angles instead of subtracting them from 180.
• D, $245$ — used $360°\mathclose{,}$ which is the total for a quadrilateral or the angles around a point.

Takeaway: Triangle angles total $180°\mathclose{;}$ quadrilateral angles total $360°\mathclose{.}$ Add your three angles at the end as a check — it costs nothing.

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Answer: C

Explanation

$AB = AC$ makes the triangle isosceles, so the two angles opposite those equal sides are equal. The sides $AB$ and $AC$ meet at $A\mathclose{,}$ so the equal angles are $B$ and $C\mathclose{.}$

The three angles total $180°\mathclose{,}$ and $A$ takes $40°\mathclose{,}$ leaving $140°$ shared equally between $B$ and $C\mathclose{:}$

$$\frac{180 - 40}{2} = \frac{140}{2} = 70$$

Check: $40 + 70 + 70 = 180$ ✓

Why each wrong option is wrong:

• A, $140$ — found the $140°$ that $B$ and $C$ share and forgot to halve it.
• B, $100$ — assumed $B$ and $C$ were $40°$ each.
• D, $40$ — that would make it equilateral, and an equilateral triangle's angles are all $60°\mathclose{,}$ not $40°\mathclose{.}$

Takeaway: Equal sides give equal opposite angles. Subtract the odd angle from 180 and halve what is left — and identify which two angles are the equal pair before you do.

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Answer: C

Explanation

Same-side interior angles (also called co-interior angles) — the pair inside the parallel lines and on the same side of the crossing line, forming a C shape — are supplementary: they add to $180°\mathclose{.}$

$$180 - 118 = 62$$

Check: $118 + 62 = 180$ ✓

It is worth keeping the three relationships apart, because all three angles exist in every one of these figures:

Why each wrong option is wrong:

• A, $242$ — subtracted from 360.
• B, $118$ — that is the corresponding or alternate angle, which is $118°\mathclose{.}$ The question asked for the co-interior one.
• D, $28$ — the gap between $118°$ and a right angle. This pair totals 180, not 90.

Takeaway: Two of the three parallel-line relationships give equal angles and one gives a total of $180°\mathclose{.}$ Identify which pair the question names before choosing.

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Section 2 — Right Triangles

Pythagoras is on the reference sheet:

$$c^2 = a^2 + b^2$$

where $c$ is the hypotenuse — the side opposite the right angle, always the longest. Putting a shorter side in $c$'s place is the standard error.

Two triangles appear so often they are worth recognising instantly. Both are on the reference sheet, but recognising them saves the algebra:

For the $30$–$60$–$90\mathclose{:}$ the shortest side faces $30°\mathclose{,}$ the $x\sqrt3$ faces $60°\mathclose{,}$ and the hypotenuse — twice the shortest — faces the right angle.

Two whole-number triples worth memorising because the test reuses them: 3–4–5 and 5–12–13, along with their multiples (6–8–10, 9–12–15).

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Answer: C

Explanation

$$c^2 = 9^2 + 12^2 = 81 + 144 = 225$$ $$c = \sqrt{225} = 15$$

This is the 3–4–5 triangle scaled by 3: $9 = 3 \times 3\mathclose{,}$ $12 = 3 \times 4\mathclose{,}$ so the hypotenuse is $3 \times 5 = 15\mathclose{.}$ Recognising it skips the arithmetic entirely.

Check the size: the hypotenuse must be longer than either leg but shorter than their sum. $12 < 15 < 21$ ✓

Why each wrong option is wrong:

• A, $3 \sqrt{7}$ — subtracted, which is what you do when the hypotenuse is given and you want a leg.
• B, $10.5$ — averaged them.
• D, $21$ — added the legs. That is the upper limit the hypotenuse can never reach, not the answer.

Takeaway: Squares add, then take the root. Learn 3–4–5 and 5–12–13 with their multiples — they appear constantly and turn the question into recognition.

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Answer: D

Explanation

The sides are in the ratio $x : x\sqrt3 : 2x\mathclose{,}$ where $x$ faces the $30°$ angle.

Here $x = 7\mathclose{,}$ so:

• opposite $30°\mathclose{:}$ $7$
• opposite $60°\mathclose{:}$ $7\sqrt3 \approx 12.1$
• hypotenuse: $2 \times 7 = 14$

The hypotenuse is twice the shortest side — the single most useful fact about this triangle.

Check with Pythagoras: $7^2 + (7\sqrt3)^2 = 49 + 147 = 196 = 14^2$ ✓

Why each wrong option is wrong:

• A, $3.5$ — halved rather than doubled. The hypotenuse is the longest side, so it cannot be smaller than a leg.
• B, $7 \sqrt{2}$ — the $45$–$45$–$90$ ratio, which belongs to the other special triangle.
• C, $7 \sqrt{3}$ — the side opposite $60°\mathclose{.}$

Takeaway: In a $30$–$60$–$90\mathclose{,}$ the hypotenuse is twice the shortest side, and the shortest side faces the $30°\mathclose{.}$ Check any answer against "the hypotenuse is the longest side".

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Answer: A

Explanation

SOH CAH TOA:

$$\sin = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan = \frac{\text{opposite}}{\text{adjacent}}$$

The right angle is at $C\mathclose{,}$ so the hypotenuse is $AB = 13\mathclose{.}$

For angle $A\mathclose{,}$ the opposite side is the one that does not touch $A$ — that is $BC = 5\mathclose{.}$

$$\sin A = \frac{5}{13}$$

The third side is $AC = \sqrt{169 - 25} = 12\mathclose{,}$ so this is the 5–12–13 triangle.

Why each wrong option is wrong:

• B, $\frac{5}{12}$ — $\tan A = \frac{5}{12}\mathclose{.}$
• C, $\frac{12}{13}$ — $\cos A = \frac{12}{13}\mathclose{,}$ using the adjacent side.
• D, $\frac{13}{5}$ — upside down. Sine and cosine are always at most 1, because the hypotenuse is the longest side. Any option greater than 1 for a sine or cosine can be eliminated immediately.

Takeaway: Identify the hypotenuse first, then which side is opposite the angle named. And remember $\sin$ and $\cos$ can never exceed 1 — it eliminates an option for free.

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Section 3 — Circles

On the reference sheet:

$$A = \pi r^2 \qquad C = 2\pi r$$

Not on it, and needed:

$$\textbf{Circle equation: } (x - h)^2 + (y - k)^2 = r^2$$

with centre $(h, k)$ and radius $r\mathclose{.}$ Two things to watch: the signs flip — $(x-3)^2$ means the centre is at $x = 3$ — and the right-hand side is $r^2\mathclose{,}$ not $r\mathclose{.}$ If it says 25, the radius is 5.

Arcs and sectors are just fractions of the whole circle. A slice with a central angle of $\theta$ degrees takes $\frac{\theta}{360}$ of everything:

$$\text{arc length} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^2$$

You do not need to memorise those two — work out the fraction and multiply.

Radians are the other unit for angles, and the reference sheet gives the one fact that converts them: a full circle is $360°\mathclose{,}$ and it is also $2\pi$ radians. Halve both:

$$180° = \pi \text{ radians}$$

To turn degrees into radians, multiply by $\frac{\pi}{180}\mathclose{;}$ to turn radians into degrees, multiply by $\frac{180}{\pi}\mathclose{.}$ Four are worth knowing on sight:

An angle of more than a full turn points the same way as whatever is left once the whole turns are taken out, so it has the same sine, cosine and tangent. $\frac{13\pi}{6}$ is one full turn, $\frac{12\pi}{6}\mathclose{,}$ plus $\frac{\pi}{6}\mathclose{.}$

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Answer: B

Explanation

Compare with $(x - h)^2 + (y - k)^2 = r^2\mathclose{.}$

$$(x - \mathbf{4})^2 + (y - (\mathbf{-3}))^2 = 25$$

The $y$ bracket reads $(y + 3)\mathclose{,}$ which is $(y - (-3))\mathclose{,}$ so $k = -3\mathclose{.}$ The signs flip: what you read is the negative of the coordinate.

Centre: $(4, -3)\mathclose{.}$

The right-hand side is $r^2 = 25\mathclose{,}$ so

$$r = \sqrt{25} = 5$$

Check: the point $(9, -3)$ is 5 units right of the centre. Substituting gives $25 + 0 = 25$ ✓

Why each wrong option is wrong:

• A — flipped the signs the wrong way and took 25 as the radius.
• C — right centre, but read $r^2$ as $r\mathclose{.}$
• D — right radius, wrong centre.

Takeaway: Signs flip for the centre; take the square root for the radius. Both mistakes are offered separately and together, so check both before choosing.

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Answer: B

Explanation

A $60°$ sector is $\frac{60}{360} = \frac16$ of the circle.

The whole circle's area is

$$\pi r^2 = \pi(6)^2 = 36\pi$$

So the sector is

$$\frac{1}{6} \times 36\pi = 6\pi$$

Why each wrong option is wrong:

• A, $2 \pi$ — the arc length, $\frac16 \times 2\pi(6) = 2\pi\mathclose{.}$ An arc is a length; a sector is an area. Check the units the question asks for.
• C, $12 \pi$ — used $2\pi r^2\mathclose{,}$ which is not a formula for anything.
• D, $36 \pi$ — the whole circle, with the fraction never applied.

Takeaway: Find the fraction $\frac{\theta}{360}\mathclose{,}$ then multiply by whichever whole-circle quantity you want — area for a sector, circumference for an arc.

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Answer: 4

Explanation

The radius is the distance from the centre to any point on the circle. Use the distance formula — which is Pythagoras in disguise, and is not on the reference sheet:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$d = \sqrt{(2 - 2)^2 + (9 - 5)^2} = \sqrt{0 + 16} = 4$$

Here the two points share an $x$-coordinate, so they sit on a vertical line and you can just count: from $y = 5$ to $y = 9$ is 4.

This is a grid-in. Enter 4.

Takeaway: The radius is the distance from the centre to any point on the circle. When two points share a coordinate, subtract the other one — no formula needed.

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Section 4 — Area, Volume and Scaling

The area and volume formulas are all on the reference sheet, so the questions are never about recall. They are about which formula, and about what happens when a shape is scaled.

Scaling is the part worth learning, because it is not obvious:

If every length is multiplied by $k\mathclose{,}$ then area is multiplied by $k^2$ and volume by $k^3\mathclose{.}$

Double every side of a cube and its surface area quadruples while its volume goes up eight times. This is the single most tested idea in this section.

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Answer: D

Explanation

From the reference sheet, $V = \pi r^2 h\mathclose{:}$

$$V = \pi (3)^2 (10) = \pi \times 9 \times 10 = 90\pi$$

Why each wrong option is wrong:

• A, $9 \pi$ — the area of the base, $\pi r^2\mathclose{,}$ never multiplied by the height.
• B, $30 \pi$ — used $r$ instead of $r^2\mathclose{.}$
• C, $60 \pi$ — that is $2\pi r h\mathclose{,}$ the curved surface area. A volume and an area are different quantities; check what the question asked for.

Takeaway: Square the radius, not the height. The formula is on the sheet — copy it before substituting rather than working from memory.

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Answer: D

Explanation

Volume depends on the cube of a length. The formula is $V = \frac43 \pi r^3\mathclose{,}$ so replacing $r$ with $3r\mathclose{:}$

$$\frac43 \pi (3r)^3 = \frac43 \pi \cdot 27 r^3 = 27 \times \left(\frac43 \pi r^3\right)$$

The volume is multiplied by $\mathbf{27}\mathclose{.}$

The $3$ inside the bracket gets cubed along with the $r$ — that is the whole question. Try it with numbers if it feels abstract: a sphere of radius 1 has volume $\frac43\pi\mathclose{;}$ radius 3 gives $36\pi\mathclose{;}$ and $36 \div \frac43 = 27$ ✓

Why each wrong option is wrong:

• A, $3$ — the factor for lengths.
• B, $6$ — doubled the 3. Nothing in the formula doubles; the only thing that happens to the 3 is that it is cubed.
• C, $9$ — the factor for areas, which is $3^2\mathclose{.}$

Takeaway: Lengths $\times k\mathclose{,}$ areas $\times k^2\mathclose{,}$ volumes $\times k^3\mathclose{.}$ Decide which kind of quantity is being asked about, then apply the right power.

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Section 5 — Mixed Practice

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Answer: 57

Explanation

Angles on a straight line add to $180°\mathclose{:}$

$$180 - 52 - 71 = 57$$

Check: $52 + 71 + 57 = 180$ ✓

This is a grid-in. Enter 57.

Takeaway: Straight line, $180°\mathclose{.}$ Around a point, $360°\mathclose{.}$ Add your answer back in as a check.

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Answer: C

Explanation

Similar means the same shape at a different size, so every length is multiplied by the same factor. Find it from the pair of sides you know:

$$k = \frac{20}{8} = 2.5$$

The smaller perimeter is $4 + 6 + 8 = 18\mathclose{.}$ Perimeter is a length, so it scales by the same $k\mathclose{:}$

$$18 \times 2.5 = 45$$

Check by scaling each side: $10, 15, 20\mathclose{,}$ which sum to 45 ✓

A gave the smaller triangle's perimeter. B scaled only one side. D matched the 20 with the shortest side of the smaller triangle, a factor of 5; the longest side corresponds to the longest.

Takeaway: Find the scale factor from a matching pair, then apply it to whatever length is asked for. Perimeter scales like a length — no squaring.

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Answer: C

Explanation

$$A = \tfrac12 bh = \tfrac12 (14)(9) = 63$$

D forgot the $\frac12$ — the most common slip with this formula. A added. B halved twice.

Takeaway: The half is part of the formula, not an optional flourish. It is on the reference sheet; copy it out before substituting.

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Answer: A

Explanation

Draw it. The ladder is the hypotenuse, the ground distance of 2.5 m is adjacent to the $65°$ angle, and the wall is opposite.

Adjacent and hypotenuse means cosine:

$$\cos 65° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2.5}{L}$$

Rearranging for $L\mathclose{:}$

$$L = \frac{2.5}{\cos 65°}$$

Sanity-check without computing anything. The ladder is the hypotenuse, so it must be longer than 2.5 m. Cosine of an acute angle is between 0 and 1, so dividing by it makes the number bigger ✓ while multiplying by it would make it smaller — which eliminates options B and D immediately.

B and D both produce something shorter than 2.5 m. C uses the opposite side, which is the wall's height, not the given distance.

Takeaway: Label hypotenuse, opposite and adjacent before choosing a ratio. Then check the size: the hypotenuse is the longest side, so any expression making it smaller is wrong.

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Answer: C

Explanation

Work back to the radius first:

$$2\pi r = 16\pi \quad\Longrightarrow\quad r = 8$$

Now the area:

$$A = \pi r^2 = \pi(8)^2 = 64\pi$$

A used $r = 4\mathclose{.}$ D treated $16$ as the radius. B applied a circumference formula.

Takeaway: Circumference to area always goes through the radius. Find $r\mathclose{,}$ then use it — never move between the two formulas directly.

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Answer: D

Explanation

An exterior angle equals the sum of the two opposite interior angles:

$$52 + 61 = 113$$

Check it the long way. The third interior angle is $180 - 113 = 67\mathclose{,}$ and the exterior angle sits on a straight line with it: $180 - 67 = 113$ ✓ Both routes agree, which is exactly why the shortcut is safe.

C gave that third interior angle — a real number in the figure, and the wrong one. A subtracted the two given angles. B averaged the two given angles where they should be added.

Takeaway: Exterior angle = the two opposite interior angles added. It saves a step, and the straight-line route confirms it.

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Answer: 73

Explanation

A quadrilateral's angles total $360°\mathclose{:}$

$$360 - 95 - 78 - 114 = 73$$

Check: $95 + 78 + 114 + 73 = 360$ ✓ Enter 73.

Using $180°$ here would give $-107\mathclose{,}$ and a negative angle should stop you at once.

Takeaway: Triangle $180°\mathclose{,}$ quadrilateral $360°\mathclose{.}$ A negative answer means you used the wrong total.

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Answer: B

Explanation

Walk around the outside. The 12 cm side the semicircle sits on is inside the shape now, so it is not part of the perimeter. What you walk is:

$$12 + 5 + 5 = 22 \text{ cm of straight edge}$$

plus the curve. The semicircle has diameter 12, so radius 6, and half a circumference is

$$\tfrac12 \times 2\pi(6) = 6\pi$$

Total: $22 + 6\pi$ cm.

A counted the joined side twice. C used a whole circumference; it is a semicircle. D used $\pi r^2\mathclose{,}$ which is an area.

Takeaway: For a composite perimeter, trace the outline with your finger. Any edge that ends up inside the shape is not part of it.

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Answer: A

Explanation

The distance between the two endpoints is the diameter:

$$d = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

The radius is half of it:

$$r = 5$$

(The centre, if a later part asks, is the midpoint $(4, 6)$ — which is option C, a point and not a length.)

B stopped at the diameter, which is the most common slip here. D added the horizontal and vertical gaps, $6 + 8\mathclose{,}$ and halved: a slanting distance comes from Pythagoras, not from adding.

Takeaway: Endpoints of a diameter give twice the radius. Compute the distance, then halve it — and use the midpoint for the centre.

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Answer: D

Explanation

A box has three pairs of matching faces:

$$4 \times 3 = 12, \qquad 4 \times 2 = 8, \qquad 3 \times 2 = 6$$

Each appears twice:

$$2(12 + 8 + 6) = 2(26) = 52$$

B gave the volume, 24 — a different quantity in different units. A added the three edge lengths. C is the commonest error: three faces counted instead of six.

Takeaway: Three distinct faces, each appearing twice. Doubling at the end is the step that gets forgotten.

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Answer: A

Explanation

A line parallel to the base makes a smaller triangle similar to the whole: triangle $ADE$ is similar to triangle $ABC\mathclose{.}$ The small triangle's side is the top piece, $AD = 4\mathclose{;}$ the whole triangle's matching side is the entire length, $AB = 4 + 6 = 10\mathclose{.}$

So the scale factor is $\frac{4}{10}\mathclose{,}$ and

$$\text{parallel line} = \frac{4}{10} \times 20 = 8$$

The trap is comparing 4 to 6. Those are the two pieces of one side; the similar triangle's side is 4 compared with the whole side, 10.

C made exactly that comparison, $\frac46 \times 20\mathclose{.}$ B used the bottom piece, $\frac{6}{10} \times 20\mathclose{.}$ D subtracted the 4 from the 20; similar figures scale by multiplying, never by subtracting.

Takeaway: In a "parallel line inside a triangle" figure, compare the small triangle's side with the whole side, never with the remaining piece.

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Answer: A

Explanation

The two acute angles of a right triangle add to $90°\mathclose{,}$ so they are complementary — and for complementary angles,

$$\sin A = \cos B$$

The reason is that the side opposite $A$ is the side adjacent to $B\mathclose{:}$ one side, two descriptions. So $\cos B = \frac{8}{17}\mathclose{,}$ with no calculation at all.

C is $\cos A\mathclose{,}$ which uses the other side: the triangle is 8–15–17. D is upside down, and greater than 1, which no cosine can be. B is $\tan A\mathclose{.}$

Takeaway: In a right triangle, $\sin$ of one acute angle equals $\cos$ of the other. The opposite side of one angle is the adjacent side of the other.

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Answer: B

Explanation

$V = \pi r^2 h\mathclose{,}$ and only $r$ changes:

$$\pi(2r)^2 h = \pi \cdot 4r^2 \cdot h = 4 \times (\pi r^2 h)$$

The volume is multiplied by 4, not 8 — because the height did not change. The radius appears squared, so doubling it multiplies by $2^2\mathclose{.}$

D is the answer to "all three dimensions doubled", where the factor would be $2^3 = 8\mathclose{.}$ Read which dimensions actually change. A is the factor for a length; the radius is squared here. C has no rule behind it: the 2 is squared, giving 4, and nothing in the formula triples it.

Takeaway: Apply the power that the changing dimension carries in the formula. $r$ is squared in a cylinder, so doubling $r$ quadruples the volume.

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Answer: B

Explanation

The radius is half the diameter: $r = 7\mathclose{.}$

$$A = \pi r^2 = 49\pi$$

A used 14 as the radius, which gives four times too much — an area error of exactly $2^2\mathclose{,}$ for the reason in the previous question. C is the circumference. D forgot to square the radius.

Takeaway: Halve the diameter before doing anything. A question that gives the diameter is testing that step.

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Answer: 25

Explanation

A diagonal cuts the rectangle into two right triangles, with the sides as legs:

$$d = \sqrt{49 + 576} = \sqrt{625} = 25$$

7–24–25 is a right-triangle triple in its own right, less famous than 3–4–5 and worth recognising. Check the size: the diagonal must be longer than either side but shorter than their sum, and $24 < 25 < 31$ ✓ Enter 25.

Takeaway: A rectangle's diagonal is the hypotenuse of a right triangle with the sides as legs. Watch for the familiar triples.

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Answer: B

Explanation

$180°$ is $\pi$ radians, so one degree is $\frac{\pi}{180}$ of a radian:

$$150 \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$$

Check the size. $150°$ is a little less than a half turn, and a half turn is $\pi\mathclose{,}$ so the answer must be a little less than $\pi\mathclose{.}$ $\frac{5\pi}{6}$ is ✓

Why each wrong option is wrong:

• A, $\frac{5 \pi}{12}$ — divided by 360 instead of 180. A full turn is $2\pi\mathclose{,}$ not $\pi\mathclose{.}$
• C, $\frac{6 \pi}{5}$ — the fraction upside down, $\frac{180}{150}\pi\mathclose{.}$ It is more than $\pi\mathclose{,}$ which would be more than a half turn.
• D, $150 \pi$ — multiplied by $\pi$ and never divided by 180.

Takeaway: $180° = \pi\mathclose{.}$ Degrees to radians: multiply by $\frac{\pi}{180}\mathclose{.}$ Then compare with a half turn, $\pi\mathclose{,}$ to see whether the size is sensible.

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Answer: B

Explanation

$\frac{13\pi}{6}$ is more than a full turn. Take the full turn out:

$$\frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}$$

So the angle points exactly where $\frac{\pi}{6}$ points, and $\frac{\pi}{6}$ is $30°\mathclose{.}$ In a $30°$–$60°$–$90°$ triangle the side opposite the $30°$ is half the hypotenuse:

$$\sin\left(\frac{13\pi}{6}\right) = \sin 30° = \frac{1}{2}$$

Why each wrong option is wrong:

• A, $- \frac{1}{2}$ — went $\frac{\pi}{6}$ back from a full turn instead of past it. That is $\frac{11\pi}{6}\mathclose{,}$ which lies below the horizontal, where sine is negative.
• C, $\frac{\sqrt{3}}{3}$ — the tangent of $30°\mathclose{:}$ opposite over adjacent, not opposite over hypotenuse.
• D, $\frac{\sqrt{3}}{2}$ — the cosine of $30°\mathclose{.}$ Sine uses the side opposite the angle, the short one.

Takeaway: Take out the whole turns first — every $2\pi$ brings you back to the start — then convert what is left to degrees and use the special triangle.

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