1. 3-1 The two conditions of similarity and the ratio of perimeters and areasExplanation
2. 3-2 Cases of similarity of triangles and the parallel lineExplanation
3. 3-3 Real-life applications of similarityExplanation
4. 3-4 The dilation and its propertiesExplanation
5. 3-5 Dilation on the coordinate planeExplanation
6. 3-6 Sides relative to an angle, and the definitions of sin, cos and tanExplanation
7. 3-7 Finding side lengths and complementary anglesExplanation
8. 3-8 Exact ratios of 30°, 45° and 60°Explanation
9. 3-9 Using the calculator and applications of trigonometryExplanation
10. 3-10 The slope of a line, parallel and perpendicular linesExplanation
11. 3-11 Equations of a line and the interceptsExplanation- Apply the two conditions of similarity and the three similarity cases of triangles, and use them in real-life measurement.
- Perform a dilation, state its properties and find image coordinates on the plane.
- Define sine, cosine and tangent in a right triangle, use complementary angles, and evaluate the ratios of 30°, 45° and 60°.
- Compute the slope of a line, test parallelism and perpendicularity, and write the equation of a line in its different forms.
🌳 Concept tree
Geometry and Measurement ├── 3-1 Similarity │ ├── the two conditions for polygons │ ├── ratio of perimeters = k, ratio of areas = k² │ ├── the three cases for triangles │ └── the line parallel to one side and its applications ├── 3-2 Dilation │ ├── centre, scale factor, properties │ └── dilation on the coordinate plane ├── 3-3 Trigonometric ratios in the right triangle │ ├── opposite, adjacent, hypotenuse │ ├── sin, cos, tan and complementary angles │ ├── exact ratios of 30°, 45°, 60° │ └── finding angles with the calculator └── 3-4 The straight line ├── slope, parallel and perpendicular lines └── forms of the equation and the intercepts
01 3-1 The two conditions of similarity and the ratio of perimeters and areas
- Two polygons are similar when corresponding angles are equal and corresponding sides are proportional — both at once.
- The common ratio of corresponding sides is the scale factor k; k = 1 gives congruent figures.
- Ratio of perimeters = k; ratio of areas = k². So from an area ratio take the square root to get k, and square k to get the area ratio.
- Any two regular polygons with the same number of sides are similar; any two circles are similar; two rectangles or two rhombi are not necessarily similar.
02 3-2 Cases of similarity of triangles and the parallel line
- AA: two equal angles (the third follows, since the angle sum is 180°).
- SSS: the three pairs of corresponding sides proportional — order the sides from smallest to largest before comparing.
- SAS: two pairs of sides proportional and the included angles equal.
- A line parallel to one side of a triangle divides the other two sides proportionally: DE ∥ BC gives AD/DB = AE/EC and AD/AB = AE/AC = DE/BC.
- The converse holds: proportional division proves parallelism.
- Corresponding altitudes, medians and angle bisectors are in the same ratio k as the sides.
03 3-3 Real-life applications of similarity
- Heights from shadows: parallel sun rays make similar triangles between object and shadow, so height/shadow is constant for all objects at the same instant.
- Indirect measurement: river width, tree height or building height using a mirror or a sighting stick, which creates two similar triangles.
- Maps, models and plans: the scale is a ratio of lengths; areas scale by its square, so a plan at 1 : 200 shows 1 cm² for 4 m² of real floor.
- Write the proportion with matching vertices in order, then cross-multiply; convert units before answering.
04 3-4 The dilation and its properties
- A dilation with centre O and scale factor k maps each point P to P′ on the line OP with OP′ = |k| × OP.
- k > 1 enlarges, 0 < k < 1 reduces; k > 0 keeps P′ on the same side of O, k < 0 puts it on the opposite side.
- The image is similar to the original: angles are preserved, lengths multiply by |k|, areas multiply by k².
- A line and its image are parallel (unless the line passes through the centre, in which case it maps onto itself).
- The centre, a point and its image are always collinear — that is how a missing centre or scale factor is recovered.
05 3-5 Dilation on the coordinate plane
- With centre at the origin: (x, y) → (kx, ky).
- Example: A(2, 3) with k = 2 gives A′(4, 6); with k = −3 it gives A′(−6, −9), on the opposite side of the origin.
- With a centre C(a, b) not at the origin, translate first: P′ = C + k(P − C), i.e. x′ = a + k(x − a) and y′ = b + k(y − b).
- Check similarity by comparing side lengths and by verifying that the ratios OA′/OA all equal |k|.
06 3-6 Sides relative to an angle, and the definitions of sin, cos and tan
- In a right triangle, for an acute angle θ: the opposite side faces θ, the adjacent side touches it, and the hypotenuse lies opposite the right angle and is the longest side.
- sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent (SOH-CAH-TOA).
- The ratios depend on the angle only, not on the size of the triangle — that is why similar triangles give the same values.
- Because the hypotenuse is longest, sinθ and cosθ are always between 0 and 1 for an acute angle, while tanθ can be any positive number.
- tanθ = sinθ/cosθ, and sin²θ + cos²θ = 1.
07 3-7 Finding side lengths and complementary angles
- Choose the ratio that links the known side with the unknown one, substitute and solve; use Pythagoras when two sides are known.
- Example: hypotenuse 10 and angle 30° → opposite = 10 sin 30° = 5, adjacent = 10 cos 30° = 5√3.
- Two acute angles in a right triangle are complementary (they sum to 90°), hence sinθ = cos(90 − θ), cosθ = sin(90 − θ) and tanθ = cot(90 − θ).
- So a ratio given for one angle can be read off for its complement — a standard one-mark conversion.
08 3-8 Exact ratios of 30°, 45° and 60°
- From an equilateral triangle of side 2 split by an altitude: sin30 = 1/2, cos30 = √3/2, tan30 = 1/√3 = √3/3; sin60 = √3/2, cos60 = 1/2, tan60 = √3.
- From an isosceles right triangle of leg 1: sin45 = cos45 = √2/2, tan45 = 1.
- Useful comparisons: as θ grows from 0° to 90°, sinθ and tanθ increase while cosθ decreases.
- Keep surd (exact) forms unless decimals are asked for; √3 ≈ 1.732 and √2 ≈ 1.414.
09 3-9 Using the calculator and applications of trigonometry
- To find a side: evaluate the ratio (make sure the calculator is in Degree mode), then multiply or divide as the equation requires.
- To find an angle: use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ on the ratio; the result is the acute angle, and rounding is stated to the required degree or minute.
- Applications: the angle of elevation looking up at a tower, the angle of depression looking down from a cliff (equal to the elevation by alternate angles), gradient of a road or ramp, and the length of a ladder against a wall.
- Draw the figure, mark the right angle and the given angle, label known sides, then select the ratio — a correct figure earns most of the method marks.
10 3-10 The slope of a line, parallel and perpendicular lines
- Slope m = (y₂ − y₁)/(x₂ − x₁) = rise over run between any two points of the line; it is the same pair whichever two points are chosen.
- Positive slope rises to the right, negative falls, zero is a horizontal line, and the slope of a vertical line x = c is undefined.
- Two non-vertical lines are parallel ⟺ their slopes are equal.
- Two lines are perpendicular ⟺ the product of their slopes is −1, i.e. one slope is the negative reciprocal of the other.
- Slope as a gradient: a road rising 3 m over 40 m has slope 3/40 = 0.075, often written as a percentage grade.
11 3-11 Equations of a line and the intercepts
- Through the origin with slope m: y = mx.
- Slope–intercept form: y = mx + b, where b is the y-intercept, the point (0, b).
- Given the slope and one point (x₁, y₁): y − y₁ = m(x − x₁) — substitute, then rearrange.
- Given two points: find m from the slope formula first, then use point–slope.
- Standard form Ax + By = C: the x-intercept is (C/A, 0) — set y = 0; the y-intercept is (0, C/B) — set x = 0.
- A line parallel to a given line keeps its slope; a perpendicular line takes the negative reciprocal — this is how most "find the equation" items are solved.
- Check the final equation by substituting the given point(s) before answering.
12 Exam traps and key takeaways
💡 High-yield traps
- Similarity needs equal angles and proportional sides; one alone is not sufficient.
- Perimeters scale by k, areas by k² — the question usually asks for the other one.
- In SAS similarity the equal angle must be the included angle.
- A dilation with negative k flips the figure through the centre; coordinates become kx, ky with the sign kept.
- Dilation keeps angles unchanged — a wrong option often claims the angle is multiplied too.
- sin, cos and tan are defined in a right triangle; identify the hypotenuse first, it is always opposite the right angle.
- sin30 = 1/2 and cos30 = √3/2; swapping them is the most common exact-value error.
- Complementary-angle questions test sinθ = cos(90 − θ): convert before looking for a triangle.
- Leaving the calculator in radian mode destroys every numerical trig answer.
- A vertical line has undefined slope, a horizontal line slope 0 — not the reverse.
- Perpendicular means negative reciprocal, not simply the opposite sign.
- In y = mx + b, b is the y-intercept; the x-intercept must be computed by setting y = 0.