3
Review: similarity, dilation, trigonometry, straight lines and standard deviation
Mathematics Prep 3 Term 1, Units 1–4, Review
GoalMix the ideas of Units 3 and 4: similarity, dilation, ratios, lines and standard deviation.
New words
similarity ratio k · the ratio of matching sides of similar figuresdilation · enlarging or reducing from a centre by a factor kslope · m = (y₂ − y₁) ÷ (x₂ − x₁) = tan θstandard deviation · σ = √(Σ(x − x̄)² ÷ n)
Explanation
Quick map of Units 3 and 4. Similar figures have equal angles and sides in the ratio k; areas are in the ratio k². A dilation about the origin with factor k sends (x, y) to (kx, ky). In a right triangle sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent, and sin 30° = 1/2, tan 45° = 1, tan 60° = √3. A line through two points has slope m = (y₂ − y₁) ÷ (x₂ − x₁); parallel lines have equal slopes and perpendicular lines have m₁m₂ = −1. And σ = √(Σ(x − x̄)² ÷ n), with a smaller σ meaning more consistent data.
Worked examples
A triangle with sides 6, 8, 10 is enlarged with ratio 3/2: sides 9, 12, 15; areas are in the ratio 9/4.
Dilation about the origin with k = 3 sends A(2, −1) to A′(6, −3).
Line through (1, 5) and (3, 1): m = −4/2 = −2, y − 5 = −2(x − 1), y = −2x + 7, standard form 2x + y = 7. A parallel line through (0, 0) is y = −2x; a perpendicular line has slope 1/2.
Standard deviation of 4, 6, 8: mean 6, squares 4, 0, 4 add to 8, σ = √(8/3) ≈ 1.63.
Class activity
Mini-exam. Without notes, do one question from each of the three review lessons, then check with your notes and mark the topics you missed.
Practice
1
Two similar quadrilaterals have a side ratio 2 : 5. A side of the smaller is 8 cm. Find the matching side of the larger.
8 × 5/2 = 20 cm.
2
A pole is 10 m away and the angle of elevation of its top is 60°. Find its height exactly.
10 × tan 60° = 10√3 ≈ 17.32 m.
3
Find the equation of the line through (0, 3) and (4, 11) in the form y = mx + b.
m = 8/4 = 2, so y = 2x + 3.
4
Find σ for 2, 4, 6, 8.
Mean 5; squares 9, 1, 1, 9 add to 20; σ = √5 ≈ 2.24.
5
Are the lines with slopes 2/3 and −3/2 parallel, perpendicular or neither?
Product = −1: perpendicular.