VertexMath · MPM2D · Free worksheet

Similar Triangles and Trigonometry Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Similar triangles: equal corresponding angles and proportional corresponding sides. Match sides by vertex order: in $\triangle ABC\sim\triangle DEF$ , $AB$ goes with $DE$ , whatever the triangles' positions on the page.
Scale factor $k$ : lengths multiply by $k$ , areas multiply by $k^2$ .
SOH CAH TOA: $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$ . Opposite and adjacent depend on which angle you use. Angles of elevation and depression are measured from the horizontal.
$\sin^{-1}x$ is the angle whose sine is $x$ . It is not $\dfrac{1}{\sin x}$ . If the unknown is in the denominator, $\cos 40^\circ=\dfrac{12}{x}$ gives $x=\dfrac{12}{\cos 40^\circ}$ .
Calculator in degree mode. Round lengths to one decimal place and angles to the nearest tenth of a degree unless told otherwise.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Similarity · Proportions · Trig Ratios · Solving Right Triangles

K1 Similar or Not?

a)
Two triangles both have angles $30^\circ$ , $60^\circ$ and $90^\circ$ . One has a hypotenuse of 10 cm, the other 14 cm. State whether they are congruent, similar but not congruent, or neither.
b)
Determine whether a triangle with sides 6, 8 and 10 is similar to a triangle with sides 9, 11 and 13.
c)
$\triangle ABC\sim\triangle DEF$ with $AB=6$ cm and $DE=9$ cm. The area of $\triangle ABC$ is $20$ cm². Find the scale factor from $\triangle ABC$ to $\triangle DEF$ , and the area of $\triangle DEF$ .

Stuck? Peek at a hint

Divide each side of one triangle by the matching side of the other: similar triangles give the same quotient every time.

K2 Proportions

In the diagram, $DE\parallel BC$ . Parts (a) and (b) refer to the diagram.

a)
Find $x$ .
b)
Find $y$ .
c)
$\triangle PQR\sim\triangle STU$ with $PQ=8$ , $QR=12$ , $PR=14$ and $ST=6$ . Find $TU$ and $SU$ .

Stuck? Peek at a hint

$\triangle ADE\sim\triangle ABC$ : compare $AD$ with the whole side $AB$ , not with $DB$ .

K3 Name the Ratios

Parts (a) and (b) refer to right triangle $ABC$ .

a)
State $\sin A$ , $\cos A$ and $\tan A$ as fractions.
b)
State $\sin B$ , $\cos B$ and $\tan B$ as fractions.
c)
In another right triangle, the side opposite $\theta$ is 9 cm and the hypotenuse is 15 cm. Determine $\tan\theta$ as a fraction in lowest terms.

Stuck? Peek at a hint

Stand at the angle: the hypotenuse faces the right angle, the opposite side faces your angle, and the adjacent side touches it.

K4 Missing Sides and Angles

a)
Find $x$ .
b)
Find $x$ .
c)
Find $\theta$ .

Stuck? Peek at a hint

Label the sides opp, adj and hyp from the marked angle, then pick the ratio that uses the two sides involved.

K5 Solve the Triangle

In $\triangle PQR$ , $\angle Q=90^\circ$ , $PQ=9$ cm and $PR=41$ cm.

a)
Determine $QR$ .
b)
Determine $\angle P$ .
c)
Determine $\angle R$ .

Stuck? Peek at a hint

Find which side is the hypotenuse first: it is the side across from the right angle.

Real World (A)

Indirect Measurement · Angles of Depression · Ramps

A1 Shadow Survey

At 3 p.m. on a sunny day in a Kingston park, a student measures heights and shadow lengths.

$\text{Object}$	Student	Flagpole	Lamp post
$\text{Height (m)}$	$1.6$	?	$4.8$
$\text{Shadow (m)}$	$2.4$	$13.5$	?

a)
Find the height of the flagpole.
b)
Find the length of the lamp post's shadow.
c)
At 5 p.m. the student's shadow is 4.0 m and the flagpole's is 22.5 m. A classmate multiplies 22.5 by the 3 p.m. ratio $\frac{1.6}{2.4}$ and claims the flagpole is 15 m tall. Is the classmate right? Give the flagpole's height.

Stuck? Peek at a hint

Keep the same order on both sides: $\dfrac{\text{height}}{\text{shadow}}=\dfrac{\text{height}}{\text{shadow}}$ .

A2 Lighthouse Watch

From the top of a lighthouse on Georgian Bay, 32 m above the water, a keeper sees a sailboat at an angle of depression of $14^\circ$ .

a)
Find the horizontal distance from the base of the lighthouse to the boat, to the nearest metre.
b)
A few minutes later, the angle of depression is $22^\circ$ . How far did the boat travel toward the lighthouse, to the nearest metre?
c)
The keeper wants the distance to a boat seen at an angle of depression of $0^\circ$ . What happens in your calculation, and what does it mean for the boat?

Stuck? Peek at a hint

Draw the horizontal line at the top of the lighthouse first; the angle of depression sits between it and the line of sight.

A3 Accessible Ramp

A school's front door is 0.75 m above the sidewalk. A local accessibility guideline says a wheelchair ramp may rise at most 1 m for every 12 m of horizontal run (a slope of 1:12).

a)
Find the shortest horizontal run allowed, and the angle of inclination of that ramp.
b)
A contractor proposes a straight ramp inclined at $8^\circ$ . Find the length of the ramp surface, and decide whether it meets the guideline.
c)
Only 6 m of sidewalk lies between the door and the street. Can a straight ramp that meets the guideline fit? What could the builder do instead?

Stuck? Peek at a hint

A 1:12 slope means $\tan\theta=\frac{1}{12}$ : rise over horizontal run, not an angle in degrees.

Think It Through (T)

Error Analysis · Counterexample · Explain Why · Build a Triangle

T1 Spot the Flaw

In right triangle $ABC$ , $\angle C=90^\circ$ , $\angle A=38^\circ$ , $AC=15$ cm and $AB=x$ . A student writes:
" $\cos38^\circ=\dfrac{15}{x}$ , so $x=15\cos38^\circ\approx11.8$ cm."

a)
Without redoing the calculation, explain how you can tell that 11.8 cm is wrong.
b)
Identify the error and find $x$ .
c)
The student then writes: " $\sin\theta=0.6$ , so $\theta=\dfrac{1}{\sin 0.6}$ ." Explain the error and find $\theta$ .

Stuck? Peek at a hint

For (a), ask which side of a right triangle is always the longest.

T2 Same Angle, Same Ratio

a)
A student claims: "All right triangles are similar, because they all have a $90^\circ$ angle." Give a counterexample.
b)
$\triangle ABC$ and $\triangle DEF$ have $\angle C=\angle F=90^\circ$ and $\angle A=\angle D=32^\circ$ , with $AB=5$ cm and $DE=12$ cm. Without a calculator, explain why $\dfrac{BC}{AB}=\dfrac{EF}{DE}$ , and name this ratio.
c)
Design a right triangle with a hypotenuse of 10 cm in which $\sin A=\cos A$ . Give its other two side lengths and $\angle A$ .

Stuck? Peek at a hint

Similar triangles need two pairs of equal angles; the right angle gives you only one.

Boss Level

Optional challenge

Crossed Wires

Two vertical poles stand on level ground. A wire runs from the top of each pole to the base of the other, and the wires cross at $P$ .

a)
The poles are 6 m and 3 m tall and stand 8 m apart. Find the height of $P$ above the ground.
b)
The same poles are moved so they stand 20 m apart. Find the new height of $P$ .
c)
A 10 m pole and a second pole give a crossing height of 4 m. Find the height of the second pole.

Stuck? Peek at a hint

Drop a perpendicular from $P$ to the ground. Each wire then forms a pair of similar triangles with one of the poles.

Answer key

Final answers only. Try first, then check.

Core Skills

a) Similar (AA) but not congruent; scale factor $1.4$
b) Not similar: $\frac{9}{6}=1.5$ but $\frac{11}{8}=1.375$
c) $k=1.5$ ; area $=45$ cm²

a) $x=3$
b) $y=10$
c) $TU=9$ , $SU=10.5$

a) $\sin A=\frac{21}{29}$ , $\cos A=\frac{20}{29}$ , $\tan A=\frac{21}{20}$
b) $\sin B=\frac{20}{29}$ , $\cos B=\frac{21}{29}$ , $\tan B=\frac{20}{21}$
c) $\tan\theta=\frac{3}{4}$ (adjacent side $=12$ cm)

a) $x\approx20.9$
b) $x\approx11.5$
c) $\theta\approx72.5^\circ$

a) $QR=40$ cm
b) $\angle P\approx77.3^\circ$
c) $\angle R\approx12.7^\circ$

Real World

a) $9$ m
b) $7.2$ m
c) No: the sun is lower at 5 p.m., so the height-to-shadow ratio has changed; the triangles are similar only at the same moment. With the 5 p.m. ratio, $22.5\times\frac{1.6}{4.0}=9$ m, the same height as before.

a) $\approx128$ m
b) $\approx49$ m (from $\approx128$ m to $\approx79$ m)
c) $\tan0^\circ=0$ , so $\frac{32}{\tan0^\circ}$ is undefined. A horizontal line of sight never meets the water, so no boat on the water is seen at $0^\circ$ ; the smaller the angle, the farther away the boat.

a) Run $=9$ m; angle $\approx4.8^\circ$
b) $\approx5.4$ m; it does not meet the guideline: $8^\circ>4.8^\circ$ (it rises 1 m for about every 7.1 m of run)
c) No: it needs 9 m of run and only 6 m is available. The builder can split the ramp into two runs joined by a landing (a switchback), or run it along the front wall of the building.

Think It Through

a) $AB$ is the hypotenuse, so it must be longer than the leg $AC=15$ cm, and $11.8<15$ .
b) The student multiplied instead of dividing: $x\cos38^\circ=15$ , so $x=\frac{15}{\cos38^\circ}\approx19.0$ cm.
c) $\sin^{-1}$ is the inverse (the angle whose sine is $0.6$ ), not the reciprocal $\frac{1}{\sin}$ . $\theta=\sin^{-1}(0.6)\approx36.9^\circ$ .

a) e.g. a 3-4-5 triangle (acute angles $\approx36.9^\circ$ and $\approx53.1^\circ$ ) and an isosceles right triangle (acute angles $45^\circ$ and $45^\circ$ ): their angles differ, so they are not similar.
b) The triangles are similar (AA), with scale factor $2.4$ : $EF=2.4\,BC$ and $DE=2.4\,AB$ , so the factor $2.4$ cancels in the ratio. The ratio is $\sin32^\circ$ ; it depends only on the angle.
c) Both legs $5\sqrt{2}\approx7.1$ cm (an isosceles right triangle); $\angle A=45^\circ$

Boss Level

Boss

a) $2$ m
b) $2$ m (the height does not depend on the distance)
c) $\frac{20}{3}\approx6.7$ m

Free cheat sheet

Similar Triangles and Trigonometry Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

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Most testedFrom the top of a $40$ m cliff, the angle of depression to a boat is $20^\circ$ . How far is the boat from the base of the cliff? (card 6)

1Congruent or similar?

SSS, SAS or ASA match	congruent: same size and shape
two pairs of equal angles (AA)	similar
all three side ratios equal	similar

TrapAdding $2$ to each side of a $3$ - $4$ - $5$ triangle gives $5$ - $6$ - $7$ : not similar, since $5/3\ne6/4$ .

TrapTwo right triangles are similar only if one more angle matches.

2Scale factor

$k=\dfrac{\text{new side}}{\text{original side}}$

lengths $\times k\qquad$ areas $\times k^2$

UseMatch sides by vertex order: in $\triangle ABC\sim\triangle DEF$ , $AB$ goes with $DE$ .

Example $k=2.5$ : a side of $4$ becomes $10$ ; an area of $6$ becomes $6(2.5)^2=37.5$ .

TrapWith $k=3$ , an area of $5$ cm $^2$ becomes $45$ cm $^2$ , not $15$ .

3Proportions and shadows

UseWrite matching sides as equal ratios, in the same order on both sides, then cross-multiply.

ExampleA $1.6$ m student casts a $2$ m shadow; a tree casts $15$ m: $h/15=1.6/2$ , so $h=12$ m.

TrapNested triangles: $AD/AB=DE/BC$ uses the whole side $AB$ , not the piece $DB$ .

TrapSame order: height/shadow $=$ height/shadow, never height/shadow $=$ shadow/height.

4SOH CAH TOA

$\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\quad\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\quad\tan\theta=\dfrac{\text{opp}}{\text{adj}}$

UseStand at $\theta$ : the opposite side doesn't touch it; the hypotenuse faces the right angle. A ratio depends only on the angle, not the triangle's size.

TrapFrom the other acute angle, opposite and adjacent swap: the vertical side is not always "opposite".

5Solving a right triangle

UseSide: write the ratio, then rearrange. Angle: use $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ . Third side: $a^2+b^2=c^2$ , $c$ the hypotenuse. Degree mode.

Example $\cos40^\circ=12/x$ : $x=12/\cos40^\circ\approx15.7$ .

TrapUnknown in the denominator: divide. $12\cos40^\circ\approx9.2$ would make the hypotenuse shorter than a leg.

Trap $\sin\theta=0.6$ gives $\theta=\sin^{-1}0.6\approx36.9^\circ$ ; $\sin^{-1}$ is not $1/\sin$ .

6Elevation and depression

UseBoth are measured from the horizontal. Depression from the top equals elevation from the bottom. Two triangles: solve one, then carry the shared side.

ExampleFrom $40$ m up, depression $20^\circ$ : $\tan20^\circ=40/d$ , so $d=40/\tan20^\circ\approx109.9$ m.

TrapA ramp slope of $1:12$ means $\tan\theta=1/12$ , so $\theta\approx4.8^\circ$ , not $12^\circ$ .

Which tool?

Two triangles with matching angles Proportion

Right triangle, two sides, need the third

a^2+b^2=c^2

Right triangle with an angle SOH CAH TOA

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