Chapter 5 — Geometry

Answer: A

Explanation

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

$$47 + 68 + x = 180$$ $$115 + x = 180$$ $$x = 65°$$

Check: $47 + 68 + 65 = 180\mathclose{.}$ Correct.

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 and stopped. 115 is a step on the way, not the answer.
• D, $245$ — subtracted from $360°\mathclose{,}$ the total for a quadrilateral.

Always add all three of your angles at the end; it takes two seconds and catches every one of these.

Takeaway: the angles of a triangle add to $180°\mathclose{.}$ Subtract the sum of the two you know, then add all three back to check.

---

Answer: H

Explanation

$AB = AC\mathclose{,}$ so the angles opposite those sides — angles $C$ and $B$ — are equal. Call each of them $x\mathclose{.}$

$$40 + x + x = 180$$ $$2x = 140$$ $$x = 70°$$

Check: $40 + 70 + 70 = 180\mathclose{.}$ Correct.

Which angles are equal? The ones opposite the equal sides. $AB$ and $AC$ both start at $A\mathclose{,}$ so the equal angles are the other two, $B$ and $C\mathclose{.}$ Getting this backwards is the usual error, and drawing the triangle makes it obvious.

Why each wrong option is wrong:

• F, $140$ — found the $140°$ left over and forgot it is shared between two angles.
• G, $100$ — treated angle $C$ as $40°$ as well.
• J, $40$ — assumed the triangle is equilateral.

Takeaway: equal sides face equal angles. Subtract the odd angle from $180°\mathclose{,}$ then halve what is left.

---

Answer: B

Explanation

The exterior angle rule:

$$\text{exterior angle} = 52 + 61 = 113°$$

Why it works: the third interior angle is $180 - 52 - 61 = 67°\mathclose{,}$ and the exterior angle sits on a straight line with it, so it is $180 - 67 = 113°\mathclose{.}$ The rule is just those two steps combined — which is why it saves time.

Why each wrong option is wrong:

• A, $67$ — gave the third interior angle, the one the exterior angle sits next to.
• D, $128$ and C, $119$ — used only one of the two given angles.

Takeaway: an exterior angle equals the sum of the two remote interior angles. If you forget the rule, find the third angle and subtract from $180°$ — same answer, one more step.

---

Answer: G

Explanation

Total interior angles:

$$(6 - 2) \times 180 = 4 \times 180 = 720°$$

The hexagon is regular, so all six angles are equal:

$$\frac{720}{6} = 120°$$

Why each wrong option is wrong:

• F, $60$ — gave the exterior angle. The exterior angles of any polygon add to $360°\mathclose{,}$ so each is $\frac{360}{6} = 60°$ — and indeed $120 + 60 = 180\mathclose{,}$ as they sit on a straight line together.
• H, $180$ — used $6 \times 180$ for the total instead of $(6 - 2) \times 180\mathclose{.}$ An angle of $180°$ is a straight line, not a corner.
• J, $720$ — gave the total for all six angles.

Takeaway: $(n-2) \times 180$ is the total. Divide by $n$ for one angle of a regular polygon. The exterior angles always total $360°\mathclose{,}$ whatever $n$ is.

---

Answer: B

Explanation

Draw both triangles. Each has a vertical object and a horizontal shadow, and the sun makes the same angle in both — so they are similar, and matching sides are in the same ratio.

Set up the proportion with height on top in both:

$$\frac{\text{flagpole height}}{\text{flagpole shadow}} = \frac{\text{post height}}{\text{post shadow}}$$

$$\frac{h}{9} = \frac{2}{1.5}$$

$$h = 9 \times \frac{2}{1.5} = 9 \times \frac{4}{3} = 12 \text{ m}$$

Sanity check: the post is taller than its shadow $(2$ against $1.5)\mathclose{,}$ so the flagpole must be taller than its 9-metre shadow. 12 is. Correct.

Why each wrong option is wrong:

• A, $18$ — multiplied the 9 by the post's height and never divided by the post's shadow.
• C, $9.5$ — added: the post is half a metre taller than its shadow, so the flagpole was made half a metre taller than its own. Similar triangles scale by a ratio, never by a difference.
• D, $6.75$ — set the proportion up upside down, which makes the flagpole shorter than its shadow and fails the sanity check.

Takeaway: similar shapes give equal ratios. Put the same measurement on top of both fractions, then check the answer is on the right side of the number you started with.

---

Section 2 — Circles

Four facts carry almost every ACT circle question:

$$\text{circumference} = 2\pi r \qquad \text{area} = \pi r^{2}$$

An arc is part of the circumference and a sector is a slice of the area. Both are the same fraction of the whole, and that fraction is $\dfrac{\text{central angle}}{360}\mathclose{:}$

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

And the equation of a circle with centre $(h, k)$ and radius $r\mathclose{:}$

$$(x - h)^2 + (y - k)^2 = r^2$$

Note the minus signs in the brackets and that the right-hand side is $r^2\mathclose{,}$ not $r\mathclose{.}$ Both are where the marks go.

---

Answer: G

Explanation

A $60°$ sector is $\frac{60}{360} = \frac{1}{6}$ of the circle.

The whole circle's area:

$$\pi r^2 = \pi \times 36 = 36\pi$$

One sixth of it:

$$\frac{1}{6} \times 36\pi = 6\pi \text{ square metres}$$

Why each wrong option is wrong:

• F, $2 \pi$ — used the arc-length formula, which gives a length, not an area. Check the units: an area needs $r^2$ in it.
• H, $12 \pi$ — used $2\pi r^2$ in place of $\pi r^2\mathclose{,}$ carrying the 2 over from the circumference formula.
• J, $36 \pi$ — gave the area of the whole circle without taking the fraction.

Takeaway: a sector is $\frac{\theta}{360}$ of the circle. Use $\pi r^2$ for area and $2\pi r$ for arc length, and let the units tell you which you need.

---

Answer: D

Explanation

The standard form is

$$(x - h)^2 + (y - k)^2 = r^2$$

with centre $(h, k)$ and radius $r\mathclose{.}$

Match it up carefully. $(x - 4)^2$ gives $h = 4\mathclose{.}$ For the $y$ term, rewrite $(y + 3)^2$ as $(y - (-3))^2\mathclose{,}$ so $k = -3\mathclose{.}$

The right-hand side is $r^2 = 25\mathclose{,}$ so $r = 5$ — take the square root.

Centre $(4, -3)\mathclose{,}$ radius 5.

Why each wrong option is wrong:

• B, centre $\left( -4, \ 3\right)\mathclose{,}$ radius $5$ — copied the signs as they appear in the brackets. The formula has a minus built in, so a $+3$ inside means $-3$ in the centre.
• C, centre $\left( 4, \ -3\right)\mathclose{,}$ radius $25$ — gave 25 as the radius. The equation gives $r^2\mathclose{.}$
• A, centre $\left( -4, \ 3\right)\mathclose{,}$ radius $25$ — both errors together.

Takeaway: the signs in the brackets flip, and the number on the right is $r^2\mathclose{.}$ Rewrite $(y+3)$ as $(y-(-3))$ if the sign flip is not automatic yet.

---

Answer: F

Explanation

Find the distance between the two points first:

$$d = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

That distance is the diameter, so the radius is half of it:

$$r = \frac{10}{2} = 5$$

The 6, 8, 10 triangle here is a scaled 3, 4, 5 — worth recognising, because it saves the arithmetic entirely.

Why each wrong option is wrong:

• G, $10$ — gave the diameter. Stopping one step early is the usual error in this question, which is exactly why it says "radius".
• H, $14$ — added the horizontal and vertical differences without squaring. That is the distance you would walk along two sides, not the straight line.
• J, $50$ — added the squares to reach 100 and halved that, never taking the square root.

Takeaway: the distance formula is Pythagoras in disguise. And read the last word — diameter or radius decides whether you halve.

---

Section 3 — Solids and Volume

The volumes worth knowing, all of which are in Chapter 8 with their reasons:

The ACT also likes scaling: what happens to area and volume when a length changes. The rule is worth more than any single formula:

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

---

Answer: C

Explanation

$$V = \pi r^2 h = \pi \times 3^2 \times 10 = \pi \times 9 \times 10 = 90\pi$$

Square the radius first, then multiply by the height. Doing it in the other order is fine too, as long as only the radius gets squared.

Why each wrong option is wrong:

• A, $30 \pi$ — forgot to square the radius.
• B, $60 \pi$ — used $2\pi r h\mathclose{,}$ which is the curved surface area, not a volume. A volume needs three lengths multiplied together, and this has only two.
• D, $900 \pi$ — squared $3 \times 10$ rather than just the 3.

Takeaway: $\pi r^2 h$ — only the radius is squared. If your expression has fewer than three lengths multiplied together, it is not a volume.

---

Answer: J

Explanation

The volume of a sphere is $\frac{4}{3}\pi r^3\mathclose{.}$ Replacing $r$ with $3r\mathclose{:}$

$$\frac{4}{3}\pi (3r)^3 = \frac{4}{3}\pi \cdot 27 r^3$$

so the volume is 27 times what it was. The bracket matters: $(3r)^3$ cubes the 3 as well as the $r\mathclose{.}$

The general rule: multiply every length by $k$ and volume multiplies by $k^3\mathclose{.}$ Here $k = 3\mathclose{,}$ so $3^3 = 27\mathclose{.}$

Why each wrong option is wrong:

• F, $3$ — assumed volume grows in proportion to length.
• G, $9$ — used $k^2\mathclose{,}$ which is the factor for area, not volume.
• H, $12$ — brought the $\frac{4}{3}$ into the scaling. Constants in the formula do not affect how it scales; only the power of $r$ does.

Takeaway: lengths scale by $k\mathclose{,}$ areas by $k^2\mathclose{,}$ volumes by $k^3\mathclose{.}$ Cube the bracket, not just the letter inside it.

---

Answer: D

Explanation

The six faces come in three pairs, one pair for each combination of two dimensions:

$$4 \times 3 = 12 \quad (\text{twice}) \qquad 4 \times 2 = 8 \quad (\text{twice}) \qquad 3 \times 2 = 6 \quad (\text{twice})$$

$$\text{surface area} = 2(12 + 8 + 6) = 2 \times 26 = 52$$

Why each wrong option is wrong:

• A, $18$ — added the dimensions rather than multiplying them in pairs.
• B, $24$ — gave the volume, $4 \times 3 \times 2\mathclose{.}$ Check the units: a surface area is in square units, and multiplying three lengths gives cubic units.
• C, $26$ — added one of each face and forgot the box has two of each.

Takeaway: a box has six faces in three matching pairs, so $2(lw + lh + wh)\mathclose{.}$ If you find yourself multiplying all three dimensions together, you are calculating a volume.

---

Section 4 — Coordinate Geometry

Three formulas connect algebra to the picture, and all three come from Pythagoras or from counting:

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

$$\text{midpoint} = \left(\frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}\right)$$

$$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$$

The midpoint is an average, so it adds; distance and slope both subtract. Mixing up which one adds is the commonest error here.

---

Answer: H

Explanation

Average each coordinate separately:

$$x: \; \frac{4 + 8}{2} = \frac{12}{2} = 6$$

$$y: \; \frac{9 + 15}{2} = \frac{24}{2} = 12$$

Midpoint $(6, 12)\mathclose{.}$

A good check: the midpoint must lie between the two endpoints in both coordinates. $6$ is between $4$ and $8\mathclose{;}$ $12$ is between $9$ and $15\mathclose{.}$ Correct.

Why each wrong option is wrong:

• F, $\left( -2, \ -3\right)$ and J, $\left( 2, \ 3\right)$ — subtracted instead of adding. Both fail the "between" check: their $y$-coordinates are nowhere near 9 to 15.
• G, $\left( 4, \ 6\right)$ — gave the differences without halving.

Takeaway: the midpoint adds and halves; distance and slope subtract. Then check your answer lies between the two endpoints.

---

Answer: D

Explanation

"5 units from the origin" means

$$x^2 + y^2 = 25$$

"3 units from the horizontal axis" means the distance from the $x$-axis is 3, so

$$y = 3 \quad \text{or} \quad y = -3$$

Take $y = 3\mathclose{:}$

$$x^2 + 9 = 25 \quad\Longrightarrow\quad x^2 = 16 \quad\Longrightarrow\quad x = \pm 4$$

That gives $(4, 3)$ and $(-4, 3)\mathclose{.}$ Now $y = -3$ gives the same two $x$ values, so $(4, -3)$ and $(-4, -3)\mathclose{.}$

Four points in total.

Why each wrong option is wrong:

• A, $1\mathclose{,}$ B, $2$ and C, $3$ — each misses some of the sign combinations. There are two independent choices, $\pm$ for $x$ and $\pm$ for $y\mathclose{,}$ so $2 \times 2 = 4\mathclose{.}$

Takeaway: "a distance from an axis" allows both sides of it, and a squared equation gives both signs. Sketch the circle and the two lines — the four crossings are then impossible to miscount.

---

Answer: G

Explanation

The points are $(-2, 1)$ and $(4, 9)\mathclose{.}$ Find the two gaps first:

$$\text{across: } 4 - (-2) = 6 \qquad \text{up: } 9 - 1 = 8$$

Then Pythagoras:

$$\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Another 6, 8, 10 triangle. The 3-4-5 family turns up constantly on the ACT.

Why each wrong option is wrong:

• F, $\sqrt{14}$ — added before squaring.
• H, $14$ — added the gaps, which is the distance walked along two sides of the right triangle rather than across it. It is always larger than the true distance.
• J, $100$ — did everything right and forgot the square root.

Takeaway: find both gaps, square, add, then root. Watch the double negative in $4 - (-2)\mathclose{;}$ that is where the arithmetic usually goes wrong.

---

Answer: C

Explanation

The scale factor comes from the pair of corresponding sides you know — the longest of each:

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

Perimeter scales by the same factor as any length:

$$(4 + 6 + 8) \times 2.5 = 18 \times 2.5 = 45$$

Why each wrong option is wrong:

• A, $18$ — gave the smaller triangle's perimeter.
• B, $30$ — replaced the longest side and left the other two unscaled.
• D, $90$ — 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 every length. Perimeter scales by $k\mathclose{;}$ area would scale by $k^2\mathclose{.}$

---

Answer: H

Explanation

Two sides are not marked, and they are found from the ones that are. The top edge is 8 and the bottom is 14, so the step goes in by $14 - 8 = 6\mathclose{.}$ The left side is 10 and the right side is 6, so the step goes up by $10 - 6 = 4\mathclose{.}$

Now cut the floor into two rectangles along the line of the step.

$$\text{lower: } 14 \times 6 = 84 \qquad \text{upper: } 8 \times 4 = 32$$

$$84 + 32 = 116$$

A second route checks it: the whole 14 by 10 rectangle is 140, and the corner that is missing is 6 by 4, which is 24. $140 - 24 = 116$ ✓

Why each wrong option is wrong:

• F, $48$ — the perimeter: $14 + 6 + 6 + 4 + 8 + 10\mathclose{.}$ That is the length of the walls, not the size of the floor.
• G, $84$ — the lower rectangle only. The upper part was left out.
• J, $164$ — $14 \times 6$ plus $8 \times 10\mathclose{.}$ The second rectangle was taken the full 10 high, so the part where the two overlap was counted twice.

Takeaway: find every missing side first, then cut the shape into rectangles that do not overlap — or take the surrounding rectangle and subtract the corner that is missing. Do both and the two answers check each other.

---

Answer: C

Explanation

Lengths are in the ratio $2 : 5\mathclose{,}$ so both dimensions scale by $\frac{5}{2}$ going from the smaller to the larger. Area is length × width, so the factor applies twice:

$$\text{area ratio} = 2^2 : 5^2 = 4 : 25$$

Why each wrong option is wrong:

• A, $2 : 5$ — used the length ratio unchanged.
• B, $4 : 5$ — squared only the 2.
• D, $8 : 125$ — cubed, which is the factor for volumes.

Takeaway: lengths scale by $k\mathclose{,}$ areas by $k^2\mathclose{,}$ volumes by $k^3\mathclose{.}$ Square both parts of the ratio.

---

Answer: F

Explanation

Three pairs of equal sides is the SSS condition, and it makes the triangles congruent — identical in every respect, not merely the same shape.

So the corresponding angles are equal and the areas are equal. A triangle's three sides fix it completely: there is no way to build two different triangles from the same three lengths.

Why each wrong option is wrong:

• G, Their corresponding angles are equal but their areas may differ — describes similar triangles, which share angles but may differ in size. Congruent is stronger.
• H, Their areas are equal but their angles may differ — equal areas do not on their own force equal angles, but here the angles are equal too.
• J, Neither the angles nor the areas need match — both do match.

Takeaway: similar means same shape, possibly different size. Congruent means identical. Three equal sides give congruence, and congruence gives you everything.

---

Answer: A

Explanation

Walk once round any polygon and you turn through one full circle, so the exterior angles always total $360°\mathclose{,}$ however many sides there are. A regular octagon has eight equal ones:

$$\frac{360}{8} = 45°$$

Check it against the interior angle. The interior angles total $(8 - 2) \times 180 = 1080°\mathclose{,}$ so each is $135°\mathclose{,}$ and $135 + 45 = 180$ — an interior angle and its exterior angle sit on a straight line together. Correct.

Why each wrong option is wrong:

• B, $135$ — the interior angle.
• C, $360$ — the total of all eight exterior angles, not one of them.
• D, $1080$ — the total of the interior angles.

Takeaway: exterior angles total $360°$ for every polygon, so each one in a regular polygon is $\frac{360}{n}\mathclose{.}$ It is often the quickest route to the interior angle as well.

---

Answer: H

Explanation

The area of a parallelogram is base times perpendicular height:

$$15 \times 8 = 120 \text{ square centimetres}$$

The 10 cm slanted side is there to be ignored. Height always means the perpendicular distance between the two parallel sides, never the sloping edge — and on the ACT that extra length is offered precisely because it is tempting.

Why each wrong option is wrong:

• F, $50$ — the perimeter, $2(15 + 10)\mathclose{.}$ That is a distance round the edge, in centimetres, not an area.
• G, $60$ — halved it, which is the triangle formula. A parallelogram is two triangles, so it does not carry the half.
• J, $150$ — used the slanted side, which overstates the area. The perpendicular height is always the shorter of the two.

Takeaway: a parallelogram is base × perpendicular height, with no half. A slanted side given alongside the height is there to be left alone.

---