Geometry (HS pathway)

Similarity and Right Triangles

Geometry

✓By the end of this lesson students will be able to identify and apply the properties of similar triangles to solve problems.
✓By the end of this lesson students will be able to understand and apply the geometric mean theorems in right triangles.
✓By the end of this lesson students will be able to define and apply the sine, cosine, and tangent ratios to find unknown side lengths and angle measures in right triangles.
✓By the end of this lesson students will be able to solve real-world problems involving angles of elevation and depression using trigonometry.

Key concepts

Similar Triangles

Two triangles are similar if their corresponding angles are congruent and their corresponding side lengths are proportional. This means one triangle is an enlargement or reduction of the other. The ratio of corresponding side lengths is called the scale factor. Criteria for proving triangles similar include Angle-Angle (AA) Similarity, Side-Angle-Side (SAS) Similarity, and Side-Side-Side (SSS) Similarity.

If ΔABC ~ ΔDEF, then ∠A≅∠D, ∠B≅∠E, ∠C≅∠F and AB/DE = BC/EF = CA/FD.

Geometric Mean Theorems (Right Triangles)

In a right triangle, if an altitude is drawn from the right angle to the hypotenuse, it divides the triangle into two smaller triangles that are similar to the original triangle and to each other. This leads to two key theorems:\n1. The altitude to the hypotenuse is the geometric mean of the two segments it divides the hypotenuse into.\n2. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

If altitude CD is drawn to hypotenuse AB in right ΔABC:\n1. CD = sqrt(AD * DB)\n2. AC = sqrt(AD * AB)\n3. BC = sqrt(DB * AB)

Trigonometric Ratios (SOH CAH TOA)

For an acute angle in a right triangle, the trigonometric ratios (sine, cosine, and tangent) relate the lengths of the sides to the angle measure. These ratios are constant for a given angle, regardless of the size of the right triangle.

For an acute angle θ in a right triangle:\nsin(θ) = Opposite / Hypotenuse\ncos(θ) = Adjacent / Hypotenuse\ntan(θ) = Opposite / Adjacent

Inverse Trigonometric Ratios

Inverse trigonometric ratios (arcsin, arccos, arctan, or sin⁻¹, cos⁻¹, tan⁻¹) are used to find the measure of an acute angle in a right triangle when the lengths of two sides are known. They are the inverse operations of the trigonometric ratios.

If sin(θ) = x, then θ = arcsin(x) or sin⁻¹(x).\nIf cos(θ) = y, then θ = arccos(y) or cos⁻¹(y).\nIf tan(θ) = z, then θ = arctan(z) or tan⁻¹(z).

Angles of Elevation and Depression

The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below the horizontal. These angles are often congruent due to parallel lines (horizontal lines) and transversals.

Key facts to remember

1Similar triangles have congruent corresponding angles and proportional corresponding sides.
2The Angle-Angle (AA) Similarity Postulate is the most common way to prove triangles are similar.
3In a right triangle, the altitude to the hypotenuse creates three similar triangles.
4SOH CAH TOA is a mnemonic for the trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
5Inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) are used to find angle measures.
6The sum of the angles in any triangle is 180 degrees.
7The Pythagorean Theorem (a² + b² = c²) applies only to right triangles, relating the lengths of the legs (a, b) to the hypotenuse (c).

Worked examples

Example 1

Given ΔABC ~ ΔXYZ. If AB = 8 cm, BC = 12 cm, AC = 16 cm, and XY = 10 cm, find the lengths of YZ and XZ.

IIdentify corresponding sides: Since ΔABC ~ ΔXYZ, AB corresponds to XY, BC corresponds to YZ, and AC corresponds to XZ.

IISet up proportions for corresponding sides: AB/XY = BC/YZ = AC/XZ.

IIISubstitute known values: 8/10 = 12/YZ = 16/XZ.

IVSolve for YZ using the first proportion: 8/10 = 12/YZ => 8 * YZ = 10 * 12 => 8 * YZ = 120 => YZ = 120 / 8 => YZ = 15 cm.

VSolve for XZ using the first proportion: 8/10 = 16/XZ => 8 * XZ = 10 * 16 => 8 * XZ = 160 => XZ = 160 / 8 => XZ = 20 cm.

Answer

YZ = 15 cm, XZ = 20 cm.

Always ensure you match corresponding sides correctly when setting up proportions.

Example 2

In right triangle PQR, with the right angle at Q, an altitude QS is drawn to the hypotenuse PR. If PS = 5 units and SR = 15 units, find the length of the altitude QS and the leg PQ. Round to the nearest tenth if necessary.

IIdentify the segments of the hypotenuse: PS = 5, SR = 15.

IIApply the Geometric Mean (Altitude) Theorem to find QS: QS = sqrt(PS * SR).

IIISubstitute values: QS = sqrt(5 * 15) = sqrt(75).

IVCalculate QS: QS ≈ 8.660... ≈ 8.7 units.

VApply the Geometric Mean (Leg) Theorem to find PQ: PQ = sqrt(PS * PR).

VIFind the length of the hypotenuse PR: PR = PS + SR = 5 + 15 = 20 units.

VIISubstitute values: PQ = sqrt(5 * 20) = sqrt(100).

VIIICalculate PQ: PQ = 10 units.

Answer

QS ≈ 8.7 units, PQ = 10 units.

Remember that the hypotenuse (PR) is the sum of its segments (PS + SR).

Example 3

A ladder leans against a wall, forming an angle of elevation of 70° with the ground. If the base of the ladder is 4 feet from the wall, how long is the ladder? Round your answer to the nearest tenth of a foot.

IDraw a diagram: Sketch a right triangle where the ladder is the hypotenuse, the wall is the opposite side, and the ground is the adjacent side.

IIIdentify knowns: Angle of elevation (θ) = 70°, Adjacent side = 4 feet.

IIIIdentify unknown: Hypotenuse (length of the ladder).

IVChoose the appropriate trigonometric ratio: Since we know the adjacent side and want to find the hypotenuse, use cosine: cos(θ) = Adjacent / Hypotenuse.

VSet up the equation: cos(70°) = 4 / Hypotenuse.

VISolve for the Hypotenuse: Hypotenuse = 4 / cos(70°).

VIICalculate the value (ensure calculator is in degree mode): Hypotenuse = 4 / 0.3420... ≈ 11.695... feet.

VIIIRound to the nearest tenth: Hypotenuse ≈ 11.7 feet.

Answer

The ladder is approximately 11.7 feet long.

Always check your calculator's mode (degrees vs. radians) before performing trigonometric calculations.

Common mistakes

✗Incorrectly identifying corresponding sides or angles when working with similar triangles.
✗Confusing the trigonometric ratios (e.g., using sine instead of cosine).
✗Forgetting to switch the calculator to 'degree' mode when solving trigonometry problems.
✗Assuming a triangle is a right triangle when it is not explicitly stated or indicated by a right angle symbol.
✗Misidentifying the angle of elevation or depression (e.g., measuring from the vertical instead of the horizontal).

Exam tips

★Always draw and label diagrams clearly for geometry problems. This helps visualize the relationships.
★For trigonometry problems, label the 'opposite', 'adjacent', and 'hypotenuse' sides relative to the angle you are working with.
★Write down the formula or ratio you are using before substituting values. This helps organize your thoughts and shows your working.
★Check your answers for reasonableness. For example, the hypotenuse should always be the longest side in a right triangle.

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