Similarity, Pythagoras and trigonometry

Foundations: angles, sides and proportionality

A triangle has three sides and three interior angles, whose sum in Euclidean plane geometry is 180°. If one angle measures 90°, the triangle is right-angled. The side opposite the right angle is called the hypotenuse; it is the longest side. The other two are legs. The hypotenuse does not depend on where we place the paper or whether the side looks sloping in the drawing.

A ratio compares two quantities through division. For lengths measured in the same unit, the ratio is a number without a unit: 6 cm/3 cm = 2. A proportion states that two ratios are equal. In 6/3 = 10/5, both pairs have the same scale factor. If a/b = c/d and b,d ≠ 0, multiplying by the denominators gives ad = bc.

Always draw before calculating. Mark the right angle, data, unknown and units. A drawing helps organise information, but does not prove that two sides are equal or that an angle is right; those properties must come from the data or an argument. If the figure is not to scale, trusting its appearance can be especially misleading.

Similarity: preserve shape, change size

Two triangles are similar when corresponding angles are equal and corresponding sides are proportional. Correspondence is essential: the side between angles A and B must be compared with the side between corresponding angles A′ and B′. Writing vertices in the correct order avoids mixing sides with different roles.

For triangles, knowing two equal corresponding angles is enough: the third also matches because of the 180° sum, and the shape is determined. Similarity can also be justified through three pairs of proportional sides, or two proportional pairs and equality of the included angle. Equality of just one angle is not enough.

If the scale factor from one triangle to another is k > 0, every length and the perimeter multiply by k. Area multiplies by k², because base and height are both lengths. A drawing enlarged with k = 3 has sides three times as long and area nine times as great. Similarity preserves shape, but not area or equal size.

Why the sum of squares appears

In a right triangle with legs a and b and hypotenuse c, Pythagoras' theorem states a² + b² = c². We can understand the equality through areas. Place four copies of this triangle inside a square of side a + b, leaving a central square of side c. The total area is (a + b)²; the sum of the four triangular areas is 4(ab/2) = 2ab.

The central area is therefore (a + b)² − 2ab = a² + b², but is also c². Hence the relationship. This explanation depends on the right angle and the construction of the central square. It does not automatically apply to an oblique triangle, where we do not have the same geometric arrangement.

To find the hypotenuse, calculate c = √(a² + b²), choosing the positive root because it is a length. To find a leg, for example a, use a = √(c² − b²). Subtraction requires c to be the hypotenuse. The converse also holds: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.

The altitude to the hypotenuse and metric relationships

Consider a triangle ABC right-angled at C. Call c the length of hypotenuse AB, b the leg AC and a the leg BC. Draw altitude CH to the hypotenuse: H belongs to AB, CH is perpendicular to AB and its length is h. The hypotenuse's parts are AH = p and HB = q; therefore c = p + q. The letters denote positive lengths in this drawing.

The two smaller triangles are similar to the original: each has a right angle and shares an acute angle with it. In triangle AHC and the original ABC, comparing corresponding sides gives b/c = p/b. Multiplying by bc gives b² = cp. Likewise, on B's side, a² = cq. Each leg squared equals the hypotenuse multiplied by that leg's projection onto it.

We can deduce the altitude relationship using the small triangle AHC: h² = b² − p². Since b² = cp and c = p + q, we get h² = p(p + q) − p² = pq. Therefore h² = pq. Adding the leg relationships, a² + b² = c(p + q) = c², recovers Pythagoras. These are not three independent rules to memorise: the same similarities connect them.

Example: if p = 9 cm and q = 16 cm, then c = 25 cm. The altitude is √(9 × 16) = 12 cm. The legs are b = √(25 × 9) = 15 cm and a = √(25 × 16) = 20 cm. Check: 15² + 20² = 625 = 25². Units also agree: the product of two lengths gives length squared.

These relationships require a right triangle and the altitude drawn to its hypotenuse. An arbitrary altitude in an oblique triangle cannot automatically be put into h² = pq. Draw the figure, identify the right angle and only then match each projection to its leg.

Sine, cosine and tangent: an angle's ratios

Choose an acute angle θ in a right triangle. The opposite leg does not touch this angle; the adjacent leg touches it and is not the hypotenuse. If you switch to the other acute angle, the two legs swap roles. The hypotenuse remains the same. This identification must come before choosing a formula.

Why do these ratios depend only on the angle, not the triangle's size? Two right triangles with the same θ also have the same third angle; they are similar. Multiplying every side by k multiplies numerator and denominator by the same factor, which cancels. Trigonometry is therefore a way of organising proportionality.

The sine and cosine of an acute angle lie between 0 and 1, because a leg is shorter than the hypotenuse. Tangent may exceed 1. The relationship tan θ = sin θ/cos θ follows from dividing the two ratios. For measuring lengths, use sides in the same unit; sine, cosine and tangent are unitless ratios.

Read a triangle through several approaches

The measurements satisfy 5² + 12² = 25 + 144 = 169 = 13². For the marked angle, sin θ = 5/13, cos θ = 12/13 and tan θ = 5/12. We do not need to calculate the angle to know these exact ratios. If we double the drawing, the sides become 10, 24 and 26; each ratio stays the same.

To recover θ from the tangent, use the inverse operation arctan: θ = arctan(5/12) ≈ 22,62°. Check the calculator mode: in this lesson numerical angles with the ° symbol are in degrees. The value arctan(5/12) in radians would be a different number representing the same opening. Rounding the ratio too early reduces the angle's precision.

Choose a relationship and examine the assumptions

If you know two sides of a right triangle and seek the third, Pythagoras is usually enough. If you know an acute angle and one side, a trigonometric ratio connects them to the side sought. If you compare two figures of the same shape, similarity may avoid calculating any angle. The choice depends on the information available, not the most recent formula you studied.

To estimate a structure's height through shadows, sunlight rays must be treatable as parallel, the ground approximately flat and measurements simultaneous. A vertical rod and vertical structure then produce similar triangles. If the ground has different slopes, the simple proportion is no longer justified.

In an angle-of-elevation measurement, the observer forms a triangle between the horizontal line, line of sight and height above the eyes. If horizontal distance is d and the angle is θ, this height difference is d tan θ. To obtain the structure's total height, add eye height. The result is an estimate conditioned by uncertainties in distance, angle and the ground model.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The hypotenuse is always the side drawn diagonally.

It is the side opposite the 90° angle, regardless of the figure's orientation.

Pythagoras holds in every triangle.

The equality a² + b² = c² requires a right triangle with c as hypotenuse.

The opposite leg always has the same name.

Opposite and adjacent depend on the chosen angle.

A scale factor of 3 multiplies area by 3.

It multiplies lengths by 3 and areas by 9.