MPM2D · Unit 6 test review

Similar Triangles and Right-Triangle Trigonometry: Test Review

Everything for the Unit 6 test on one page: the key facts from all 3 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Similar Triangles

Similar triangles: corresponding angles are equal and corresponding sides are proportional. In $△ A BC \sim △ D EF$ the vertices are listed in matching order.
Scale factor: $k = \frac{side in the new triangle}{matching side in the original} .$ Every side is multiplied by $k$ ; $k = 1$ means the triangles are congruent.
Tests for similarity: AA (two pairs of equal angles), SSS similarity (all three side ratios equal), SAS similarity (two side ratios equal and the angles between those sides equal).
Congruent triangles are similar with scale factor 1. Similar triangles are congruent only when the scale factor is 1.
To find a missing side, match sides by the angles they are opposite, write a proportion with the ratios in the same order, and solve.
If lengths are scaled by $k$ , perimeters are scaled by $k$ and areas by $k^{2}$ .

The Primary Trigonometric Ratios

The hypotenuse is the side across from the right angle and is the longest side. Opposite and adjacent are named from the reference angle and swap when the reference angle changes.
$sin θ = \frac{opp}{hyp}$ , $cos θ = \frac{adj}{hyp}$ , $tan θ = \frac{opp}{adj}$ (SOH CAH TOA).
The ratios depend only on the angle, because all right triangles with the same acute angle are similar (AA).
Finding a side: unknown on top, multiply (e.g. $x = 15.0 sin 2 8^{\circ}$ ); unknown on the bottom, divide (e.g. $x = \frac{8.0}{cos 5 2 ^{\circ}}$ ).
Finding an angle: $θ = sin^{- 1} (\frac{opp}{hyp})$ , $cos^{- 1} (\frac{adj}{hyp})$ or $tan^{- 1} (\frac{opp}{adj})$ . The two acute angles add to $9 0^{\circ}$ .
Use degree mode (check: $sin 3 0^{\circ} = 0.5$ ) and keep unrounded values until the final answer.

Solving Right-Triangle Problems

Angle of elevation: measured up from the horizontal to the line of sight. Angle of depression: measured down from the horizontal to the line of sight.
The angle of depression from P to Q equals the angle of elevation from Q to P (alternate angles between parallel horizontal lines).
In a right triangle: two sides give the third by $a^{2} + b^{2} = c^{2}$ ; a side and an acute angle give another side by SOH CAH TOA; two sides give an angle by an inverse ratio; the acute angles add to $9 0^{\circ}$ .
When an angle is measured at eye level, add the eye height to the height found from the triangle.
In two-triangle problems, solve the triangle with enough information first, find the shared side, and carry it unrounded into the second triangle. If neither triangle can be solved alone, write an equation from each and solve the system.

Where marks get lost

Pairing sides by where they appear on the page instead of by the angles they are opposite. When one triangle is turned or flipped, the bottom side of one is often not the match for the bottom side of the other. (Similar Triangles)
Flipping one ratio in the proportion, for example $\frac{h}{11.2} = \frac{2.4}{1.8}$ . Both ratios must compare the same two kinds of length in the same order. (Similar Triangles)
Thinking that triangles with equal angles must be congruent. Equal angles (AA) prove similarity only; the sizes can be different. (Similar Triangles)
Adding the same number to each side and calling the result similar. Similar triangles come from multiplying every side by the same scale factor. (Similar Triangles)
Scaling an area by $k$ instead of $k^{2}$ . A model at scale $1 : 50$ has $\frac{1}{2500}$ of the real area, not $\frac{1}{50}$ . (Similar Triangles)
Leaving the calculator in radian mode. If $sin 30$ does not give $0.5$ , switch to degrees before doing anything else. (The Primary Trigonometric Ratios)
Labelling opposite and adjacent from the wrong angle (often the right angle), or calling the hypotenuse the adjacent side. Opposite and adjacent are always legs, named from the acute reference angle. (The Primary Trigonometric Ratios)
Multiplying when the unknown is in the denominator: from $cos 5 2^{\circ} = \frac{8.0}{x}$ , writing $x = 8.0 cos 5 2^{\circ}$ instead of $x = \frac{8.0}{cos 5 2 ^{\circ}}$ . A hypotenuse that comes out shorter than a leg is the warning sign. (The Primary Trigonometric Ratios)
Reading $sin^{- 1} x$ as $\frac{1}{sin x}$ . The inverse sine takes a ratio and gives back an angle. (The Primary Trigonometric Ratios)
Rounding a ratio early, such as using $sin 2 8^{\circ} \approx 0.47$ , which can change the final rounded answer. (The Primary Trigonometric Ratios)
Measuring the angle of depression from the vertical (the cliff or wall) instead of from the horizontal, which puts the angle in the wrong corner of the triangle. (Solving Right-Triangle Problems)
Forgetting to add the observer's eye height when the angle was measured with a clinometer held at eye level. (Solving Right-Triangle Problems)
Rounding the shared side in a two-triangle problem (for example using 27.6 instead of 27.608…), which can change the final rounded answer. (Solving Right-Triangle Problems)
Using the distance between two observers as if it were the distance to the base. In the tower problem the farther person is $x + 50$ m from the base, not $50$ m. (Solving Right-Triangle Problems)
Applying SOH CAH TOA to a triangle with no right angle. The ratios as defined here need a right triangle; for an acute triangle, use the sine law or the cosine law (the next unit). (Solving Right-Triangle Problems)

Practice test

18 questions from across the unit. Each one shows the lesson it comes from.

Similar Triangles
1
Which information is enough, by itself, to prove that two triangles are similar?
Similar Triangles
2
$△ A BC \sim △ D EF$ with $A B = 5$ cm, $BC = 7$ cm and $D E = 12.5$ cm. Find $EF$ in centimetres.
cm
Similar Triangles
3
A triangle with sides 4, 6 and 8 cm is enlarged to a similar triangle with sides 10, 15 and 20 cm. What is the scale factor?
Similar Triangles
4
Put the steps for finding a missing side in similar triangles in order.
1. 1Write a proportion using a known pair of corresponding sides and the pair that contains the unknown
2. 2Check that the larger triangle has the larger side
3. 3Solve the proportion for the unknown side
4. 4Show the triangles are similar (for example by AA) and match the corresponding vertices
Similar Triangles
5
In $△ A BC$ , point $D$ is on $A B$ and point $E$ is on $A C$ with $D E$ parallel to $BC$ . $A D = 4.0$ m, $A B = 10.0$ m and $D E = 3.2$ m. Find $BC$ in metres.
m
Similar Triangles
6
Two similar triangles have a scale factor of 3. The smaller triangle has an area of 10 square centimetres. What is the area of the larger triangle, in square centimetres?
cm²
The Primary Trigonometric Ratios
7
In right triangle $PQR$ , $∠ Q = 9 0^{\circ}$ . Which side is adjacent to $∠ P$ ?
The Primary Trigonometric Ratios
8
Right triangle 1 has a 35° angle and a hypotenuse of 10 cm. Right triangle 2 has a 35° angle and a hypotenuse of 30 cm. For the 35° angle, how does opposite ÷ hypotenuse compare in the two triangles?
The Primary Trigonometric Ratios
9
In $△ A BC$ , $∠ C = 9 0^{\circ}$ , $∠ A = 2 5^{\circ}$ and $BC = 6.0$ cm. Find $A C$ , to one decimal place.
cm
The Primary Trigonometric Ratios
10
In a right triangle, the side opposite angle A is 3 and the side adjacent to angle A is 4. What is tan A?
The Primary Trigonometric Ratios
11
Put the steps for finding a missing side in a right triangle in order.
1. 1Round the final answer to the required accuracy
2. 2Solve for x by multiplying or dividing
3. 3Write the equation, such as sin 28° = x ÷ 15.0
4. 4Label the opposite, adjacent and hypotenuse from the given acute angle
5. 5Choose the ratio (SOH, CAH or TOA) that uses the known side and the unknown side
The Primary Trigonometric Ratios
12
In a right triangle, cos A = 0.6. Find angle A to the nearest degree.
°
Solving Right-Triangle Problems
13
An angle of depression is measured between the line of sight and which line?
Solving Right-Triangle Problems
14
A boat is 100 m from the base of a vertical cliff that is 60 m high. Find the angle of depression from the top of the cliff to the boat, to the nearest degree.
°
Solving Right-Triangle Problems
15
A ship sails 18.0 km east, then 7.5 km north. How far is it from its starting point, in kilometres? Round to one decimal place.
km
Solving Right-Triangle Problems
16
In a right triangle you know both legs and no acute angle. Which tool finds the hypotenuse in one step?
Solving Right-Triangle Problems
17
From a window $15.0$ m above level ground, the angle of elevation to the top of a tower is $3 5^{\circ}$ and the angle of depression to its base is $2 2^{\circ}$ . How tall is the tower? Round to one decimal place.
m
Solving Right-Triangle Problems
18
A ramp rises 0.5 m over a horizontal distance of 6.0 m. What angle does the ramp make with the ground? Round to the nearest degree.
°

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…