• 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.

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

• 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.

💡 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.