Math: Angles, Triangles and Similarity

This is Unit Thirteen, part two, of ACT Preparation. Part one closed on what each quadrilateral guarantees; this part turns to triangles of the same shape, to right triangles, and to what scaling does. The habit carries over: name the fact behind every step. Here a second habit joins it: match sides by the angles they face. By the end you will be able to solve similar triangles for a side. To measure a height with shadows. To use the Pythagorean theorem and its triples. To find the sides of the two special right triangles. And to apply a scale factor to areas and volumes.

Three entries on similar triangles, from proof to shadows.

Same shape, different size.

ACT 1219 Similar Triangles

Similar triangles have equal corresponding angles and proportional corresponding sides, and two pairs of equal angles, the angle-angle test, prove similarity. Triangles with sides three, four, five and six, eight, ten are similar, with a ratio of two. Match sides by the angles they face, not by position on the page.

A missing side, step by step.

ACT 1220 Solving Similar Triangles for a Side

Confirm similarity and match the vertices, write a proportion with corresponding sides in the same order, then cross-multiply. With D E parallel to B C, triangles A D E and A B C share angle A and are similar; A D is four, A B twelve and D E six, so four over twelve equals six over B C, and B C is eighteen. Use whole sides: the leftover piece D B, eight, corresponds to nothing in the small triangle.

Measuring what cannot be reached.

ACT 1221 Similar Triangles in the World

Sketch the two triangles, often objects and their shadows, argue similarity, since the sun's rays meet the ground at one angle and both objects stand upright, then set height over shadow equal to height over shadow. A 1.8-meter post with a 2.4-meter shadow and a tree with a sixteen-meter shadow give a tree twelve meters tall. Keep the same order on both sides; mixing them gives a different, wrong height.

Six entries on right triangles: the Pythagorean theorem and the two special triangles.

The Pythagorean theorem.

ACT 1222 The Pythagorean Theorem and Its Converse

In a right triangle with legs a and b and hypotenuse c, a squared plus b squared equals c squared, and conversely, a triangle whose sides fit is right. Legs six and eight give c squared equals one hundred, so c is ten. The hypotenuse, opposite the right angle, is the longest side.

Whole-number triples.

ACT 1223 Pythagorean Triples

A Pythagorean triple is three whole numbers that fit the theorem. Three, four, five and five, twelve, thirteen appear often, with their multiples: nine, twelve, fifteen is three, four, five times three. Recognizing a triple saves the arithmetic.

Using the theorem, step by step.

ACT 1224 Using the Pythagorean Theorem

Identify the hypotenuse, substitute, adding squares for a missing hypotenuse or subtracting for a missing leg, then take the root, checking for a triple. A thirteen-foot ladder with its foot five feet from the wall reaches twelve feet, the five, twelve, thirteen triple. Sides seven, nine and twelve give 130, not 144, so that triangle is not right. For a missing leg, subtract: adding gives about 13.9 feet, longer than the ladder itself.

The 45-45-90 triangle.

ACT 1225 The 45-45-90 Triangle

In a 45-45-90 triangle the legs are equal and the hypotenuse is a leg times root two. Legs of seven give a hypotenuse of seven root two, and a square with side five has a diagonal of five root two. From the hypotenuse, divide by root two: ten becomes five root two.

The 30-60-90 triangle.

ACT 1226 The 30-60-90 Triangle

In a 30-60-90 triangle, the hypotenuse is twice the shorter leg, which faces the thirty-degree angle, and the longer leg is the shorter leg times root three. A shorter leg of four gives a hypotenuse of eight and a longer leg of four root three. Find the shorter leg first.

Special triangles, step by step.

ACT 1227 Special Right Triangle Sides

Identify the triangle: a square cut by its diagonal gives 45-45-90, and an equilateral triangle cut by its height gives 30-60-90. Find the base side, then multiply by the ratio, rationalizing if needed. An equilateral triangle with sides of twelve has a height of six root three; a 45-45-90 triangle with hypotenuse nine has legs of nine root two over two. The shorter leg is always opposite thirty degrees.

Three entries on scale factors, and the whole unit in one figure.

What scaling does.

ACT 1228 Scale Factor for Area and Volume

When every length is multiplied by a scale factor k, areas are multiplied by k squared and volumes by k cubed. Doubling every edge of a box multiplies its surface area by four and its volume by eight. The course derives this from the area and volume formulas: each length factor carries its own k.

Applying a scale factor, step by step.

ACT 1229 Applying a Scale Factor

Find k from corresponding lengths, new over old; decide whether the quantity is a length, an area or a volume; and multiply by k, k squared or k cubed. Similar triangles with sides four and ten have k equal to 2.5, so an area of twelve becomes seventy-five. Cubes with volumes eight and 125 have edges in the ratio two to five, the cube root. Applying k to an area gives thirty, less than half the true seventy-five.

The whole unit in one figure.

ACT 1230 One Figure, Five Facts

Plane geometry runs on a short list of facts, and every step uses one. Two parallel streets 300 meters apart are crossed by a diagonal road meeting the northern street at sixty degrees, and a footpath runs straight south from A on the northern street to B on the southern. The road reaches the southern street at C, where alternate interior angles make the angle sixty degrees. Triangle A B C has a right angle at B and sixty at C, so thirty at A: a 30-60-90 triangle. The path, 300 meters, faces the sixty-degree angle, so it is the longer leg; the shorter leg B C is one hundred root three, about 173 meters; and the road, the hypotenuse, is two hundred root three, about 346. Name the fact behind each step, match sides by the angles they face, and trust the given measures over the drawing.

Look at what you can now do. You can prove triangles similar by two angles and solve for a side with whole sides. You can measure a tree from its shadow. You can use the Pythagorean theorem, subtracting for a leg, and spot the common triples. You can find every side of a special right triangle from its base side. And you can apply k to lengths, k squared to areas and k cubed to volumes. Unit fourteen measures figures, perimeter, area, volume and circles, and turns to trigonometry.