Similarity and Right Triangle Trigonometry
Summary
This chapter develops proportional reasoning through similarity transformations and similarity criteria for triangles, then introduces the Pythagorean theorem and trigonometric ratios for solving right triangles. You'll learn how similar figures relate through scale factors, apply the Pythagorean theorem to find distances, and use sine, cosine, and tangent to solve real-world problems involving angles of elevation, angles of depression, and indirect measurement.
Concepts Covered
This chapter covers the following 20 concepts from the learning graph:
- Similar Figures
- Scale Factor
- Similarity Ratio
- AA Similarity
- SAS Similarity
- SSS Similarity
- Proportional Segments
- Geometric Mean
- Altitude to Hypotenuse
- Pythagorean Theorem
- Pythagorean Triple
- Converse Pythagorean Theorem
- Distance in Coordinate Plane
- Sine Ratio
- Cosine Ratio
- Tangent Ratio
- Inverse Trigonometric Functions
- Angle of Elevation
- Angle of Depression
- Indirect Measurement
Prerequisites
This chapter builds on concepts from:
- Chapter 5: Coordinate Geometry and Lines
- Chapter 6: Transformations and Congruence
- Chapter 7: Triangle Congruence and Properties
Introduction
Have you ever wondered how surveyors measure the height of a mountain without climbing it? Or how architects create scale models of buildings? The answer lies in similarity and trigonometry—two powerful mathematical tools that allow us to work with shapes that have the same form but different sizes, and to find unknown measurements using angles and ratios.
In this chapter, you'll discover how to determine if two figures are similar, use scale factors to solve real-world problems, and unlock the secrets of right triangles through the Pythagorean theorem and trigonometric ratios. By the end, you'll be able to measure the height of your school building using just a measuring tape and a sunny day!
Understanding Similar Figures
Two figures are similar if they have the same shape but not necessarily the same size. Think of a photograph that you enlarge or reduce—the image remains the same, just at a different scale.
Similar figures have two important properties:
- Corresponding angles are congruent (equal in measure)
- Corresponding sides are proportional (their ratios are equal)
Diagram: Similar Triangles Comparison
Similar Triangles Comparison
Type: diagram
Learning Objective: Students will understand the visual relationship between similar triangles, recognizing that corresponding angles are equal and corresponding sides are proportional (Bloom's: Understanding).
Purpose: Illustrate two similar triangles side-by-side to show angle congruence and side proportionality.
Components to show: - Two triangles: △ABC (smaller) and △DEF (larger) - △ABC with sides: AB = 3 cm, BC = 4 cm, AC = 5 cm - △DEF with sides: DE = 6 cm, EF = 8 cm, DF = 10 cm - Angle markings showing: ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F - Color-coded corresponding angles (red, blue, green) - Arrows connecting corresponding vertices
Annotations: - Label showing scale factor: "Scale factor = 2" - Ratio expressions: "DE/AB = 6/3 = 2", "EF/BC = 8/4 = 2", "DF/AC = 10/5 = 2" - Note: "All ratios equal → Triangles are similar"
Style: Clean geometric diagram with clear labels and color coding Color scheme: Triangles outlined in black, angles marked in red/blue/green, ratios in purple
Implementation: Can be created using p5.js, GeoGebra, or static diagram tool
Scale Factor
The scale factor is the ratio of corresponding side lengths between two similar figures. If the scale factor is greater than 1, the image is an enlargement. If it's between 0 and 1, the image is a reduction.
Scale Factor Formula
\(k = \frac{\text{length in new figure}}{\text{length in original figure}}\)
where:
- \(k\) is the scale factor
- The numerator is a side length from the larger or scaled figure
- The denominator is the corresponding side length from the original figure
For example, if a 4-inch photograph is enlarged to 10 inches, the scale factor is:
\(k = \frac{10}{4} = 2.5\)
Drawing: Scale Factor Explorer
Scale Factor Explorer MicroSim
Type: microsim
Learning Objective: Students will apply scale factor concepts to create similar figures and explore how changing the scale factor affects dimensions while preserving shape (Bloom's: Applying).
Purpose: Allow students to interactively adjust scale factors and see how they affect similar figures.
Canvas layout (800x500px): - Top area (800x400): Drawing area showing original and scaled triangles - Bottom area (800x100): Control panel with sliders and information display
Visual elements: - Original triangle (blue) on left side with labeled dimensions - Scaled triangle (orange) on right side with labeled dimensions - Grid background for reference - Midpoint line separating original and scaled versions - Labels showing measurements on each side
Interactive controls: - Slider: Scale factor (range: 0.5 to 3.0, default: 1.5, step: 0.1) - Display: "Scale Factor: k = [value]" - Display: "Original area: [value] sq units" - Display: "Scaled area: [value] sq units" - Display: "Area ratio: [value]" - Reset button
Default parameters: - Original triangle: base = 4, height = 3 - Initial scale factor: 1.5 - Background color: light gray grid
Behavior: - When slider moves, scaled triangle updates in real-time - All dimensions of scaled triangle update proportionally - Area calculations update automatically - Show relationship: scaled area = k² × original area - Corresponding angles remain equal (show angle measures)
Educational features: - Highlight corresponding sides in matching colors when mouse hovers - Display ratio of corresponding sides - Show that area scales by k², not k
Implementation: p5.js with createSlider() for controls Canvas positioning: Controls at bottom for easy access
Triangle Similarity Criteria
Just as we had shortcuts for proving triangle congruence (SSS, SAS, ASA), we have similar shortcuts for proving triangles are similar. You don't need to check all angles and all sides—just certain combinations!
The three triangle similarity criteria are:
- AA (Angle-Angle) Similarity
- SAS (Side-Angle-Side) Similarity
- SSS (Side-Side-Side) Similarity
AA Similarity
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Why does this work? Remember that the sum of angles in any triangle is 180°. If two angles match, the third angle must also match!
Diagram: AA Similarity Proof
AA Similarity Proof
Type: diagram
Learning Objective: Students will analyze why two congruent angles are sufficient to prove triangle similarity, connecting angle sum properties to similarity criteria (Bloom's: Analyzing).
Purpose: Show two triangles with two pairs of congruent angles and demonstrate why the third angle must also be congruent.
Components: - Triangle PQR with angles: ∠P = 50°, ∠Q = 70°, ∠R = 60° - Triangle XYZ with angles: ∠X = 50°, ∠Y = 70°, ∠Z = ? - Color-coded angle marks showing congruent angles - Calculation bubble showing: "∠Z = 180° - 50° - 70° = 60°" - Checkmark indicating ∠R ≅ ∠Z
Annotations: - "Given: ∠P ≅ ∠X and ∠Q ≅ ∠Y" - "By angle sum: ∠R = 60° and ∠Z = 60°" - "Therefore: ∠R ≅ ∠Z" - "Conclusion: △PQR ~ △XYZ by AA"
Visual style: Side-by-side triangles with clear angle markings Color scheme: Matching angles in same colors (50° in red, 70° in blue, 60° in green)
Implementation: Static diagram or interactive p5.js visualization
SAS Similarity
If an angle of one triangle is congruent to an angle of another triangle, and the sides including those angles are proportional, then the triangles are similar.
This is similar to SAS congruence, but instead of sides being equal, they must be proportional with the same ratio.
SSS Similarity
If the corresponding sides of two triangles are proportional (all three ratios are equal), then the triangles are similar.
Comparison Table: Congruence vs. Similarity
Here's how the criteria compare:
| Criterion | Congruence | Similarity |
|---|---|---|
| SSS | Three sides equal | Three sides proportional (same ratio) |
| SAS | Two sides equal, included angle equal | Two sides proportional, included angle equal |
| ASA | Two angles equal, included side equal | Not used (AA is sufficient) |
| AAS | Two angles equal, non-included side equal | Not used (AA is sufficient) |
| AA | Not sufficient for congruence | Two angles equal → Similar |
Drawing: Triangle Similarity Tester
Triangle Similarity Tester MicroSim
Type: microsim
Learning Objective: Students will evaluate different triangle configurations to determine which similarity criterion applies, strengthening their ability to identify and justify similarity relationships (Bloom's: Evaluating).
Purpose: Allow students to test whether two triangles are similar by checking different criteria.
Canvas layout (850x600px): - Left section (400x500): First triangle with adjustable parameters - Right section (400x500): Second triangle with adjustable parameters - Bottom section (850x100): Control panel and results display
Visual elements: - Two triangles displayed with all side lengths and angle measures labeled - Color highlighting for corresponding parts - Visual indicators showing which elements are being compared - Similarity verdict display with criterion used
Interactive controls: - Triangle 1 sliders: - Side 1 length (range: 2-10, default: 4) - Side 2 length (range: 2-10, default: 6) - Included angle (range: 20°-140°, default: 60°) - Triangle 2 sliders: - Side 1 length (range: 2-10, default: 6) - Side 2 length (range: 2-10, default: 9) - Included angle (range: 20°-140°, default: 60°) - Button: "Check Similarity" - Dropdown: "Test method" (AA, SAS, SSS, or Auto-detect)
Behavior: - Triangles update in real-time as sliders move - Calculate third side using law of cosines - Calculate all angles - When "Check Similarity" clicked: - Compare angles (for AA) - Calculate side ratios (for SSS and SAS) - Display which criterion applies (if any) - Highlight corresponding parts - Show scale factor if similar
Results display: - "Triangles ARE similar by [criterion]" or "Triangles are NOT similar" - If similar: "Scale factor k = [value]" - Show all angle measures and side ratios - Explain which criterion was satisfied
Educational features: - Include "Challenge" button that generates random triangle pairs - Track success rate - Provide hints if students are stuck
Implementation: p5.js with geometric calculations Include law of cosines: c² = a² + b² - 2ab cos(C)
Proportional Segments and Geometric Mean
When we work with similar figures, we often encounter proportional segments—line segments whose lengths have equal ratios.
Side-Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
Diagram: Side-Splitter Theorem
Side-Splitter Theorem Illustration
Type: diagram
Learning Objective: Students will understand how parallel lines create proportional segments in triangles, applying the Side-Splitter Theorem to solve for unknown lengths (Bloom's: Understanding, Applying).
Purpose: Visualize the Side-Splitter Theorem with a triangle and a parallel line creating proportional segments.
Components: - Triangle ABC with base BC - Point D on side AB, point E on side AC - Line segment DE parallel to BC (marked with arrows) - Measurements: AD = 3, DB = 5, AE = x, EC = 8 - Proportion equation displayed: AD/DB = AE/EC
Visual elements: - Parallel markings (arrows) on DE and BC - Color-coded segments: AD and DB in different shades of blue, AE and EC in different shades of orange - Dashed line showing the parallel relationship - Solution bubble: "3/5 = x/8, so x = 4.8"
Annotations: - "If DE ∥ BC, then AD/DB = AE/EC" - Label each segment length - Show work for solving the proportion
Style: Clear geometric diagram with emphasis on proportional relationships Color scheme: Blue for left side segments, orange for right side segments
Implementation: Static diagram or GeoGebra interactive
Geometric Mean
The geometric mean of two numbers is the positive square root of their product. For two positive numbers \(a\) and \(b\):
Geometric Mean Formula
\(\text{geometric mean} = \sqrt{a \cdot b}\)
where:
- \(a\) and \(b\) are positive numbers
- The result is always between \(a\) and \(b\) (if \(a < b\))
For example, the geometric mean of 4 and 9 is:
\(\sqrt{4 \cdot 9} = \sqrt{36} = 6\)
The geometric mean appears frequently in right triangles, especially when dealing with altitude to the hypotenuse.
Altitude to Hypotenuse Theorem
In a right triangle, when you draw an altitude from the right angle to the hypotenuse, it creates three similar triangles. This altitude is the geometric mean of the two segments of the hypotenuse.
Diagram: Altitude to Hypotenuse
Altitude to Hypotenuse and Three Similar Triangles
Type: diagram
Learning Objective: Students will analyze the relationship between a right triangle's altitude to the hypotenuse and the resulting similar triangles, applying geometric mean concepts (Bloom's: Analyzing, Applying).
Purpose: Show how an altitude to the hypotenuse of a right triangle creates three similar triangles and establishes geometric mean relationships.
Components: - Right triangle ABC with right angle at C - Hypotenuse AB - Altitude CD from C perpendicular to AB, meeting at point D - Three triangles highlighted in different colors: 1. Original triangle ABC (outline) 2. Small triangle ACD (light blue fill) 3. Small triangle CBD (light orange fill) - Segments labeled: AD = a, DB = b, CD = h
Mathematical relationships displayed: - h² = a · b (altitude is geometric mean) - △ABC ~ △ACD ~ △CBD (similarity statement) - Right angle markings at C and D
Visual elements: - Color coding for the three similar triangles - Angle marks showing corresponding angles - Formula box: "CD = √(AD · DB)" - Example with numbers: If AD = 4 and DB = 9, then CD = √(4·9) = 6
Annotations: - "Altitude creates two smaller triangles similar to the original" - "The altitude is the geometric mean of the hypotenuse segments" - Label each similar triangle separately
Style: Multi-layer diagram showing overlapping triangles Color scheme: Original in black outline, smaller triangles in translucent blue and orange
Implementation: Interactive p5.js showing three triangles separately and combined
The Pythagorean Theorem
One of the most famous theorems in all of mathematics! The Pythagorean theorem relates the three sides of a right triangle.
In any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
Pythagorean Theorem Formula
\(a^2 + b^2 = c^2\)
where:
- \(a\) and \(b\) are the lengths of the two legs (sides forming the right angle)
- \(c\) is the length of the hypotenuse (side opposite the right angle)
- The order of \(a\) and \(b\) doesn't matter
This theorem lets you find any side of a right triangle if you know the other two sides.
Example: If a right triangle has legs of length 3 and 4, what is the hypotenuse?
\(3^2 + 4^2 = c^2\)
\(9 + 16 = c^2\)
\(25 = c^2\)
\(c = 5\)
Drawing: Pythagorean Theorem Visual Proof
Pythagorean Theorem Visual Proof MicroSim
Type: microsim
Learning Objective: Students will understand why the Pythagorean theorem is true through visual area relationships, deepening conceptual understanding beyond memorization (Bloom's: Understanding, Analyzing).
Purpose: Provide an interactive visual proof of the Pythagorean theorem using area squares.
Canvas layout (700x700px): - Central area: Right triangle with squares drawn on each side - Bottom area: Controls and area calculations
Visual elements: - Right triangle with legs a and b, hypotenuse c - Square drawn on leg a (colored blue) with area a² - Square drawn on leg b (colored green) with area b² - Square drawn on hypotenuse c (colored red) with area c² - Grid overlay showing unit squares for counting - Area labels on each square
Interactive controls: - Slider: Length of leg a (range: 3-10, default: 3, step: 1) - Slider: Length of leg b (range: 3-10, default: 4, step: 1) - Button: "Animate Proof" - shows squares rearranging visually - Display: "a² = [value]", "b² = [value]", "c² = [value]" - Display: "a² + b² = [sum]" - Checkbox: "Show unit squares" (shows grid for counting)
Behavior: - Right triangle updates when sliders change - All three squares update with triangle - Calculate and display all areas - "Animate Proof" button shows visual rearrangement demonstrating a² + b² = c² - Verify equation: highlight in green if equal, red if not equal
Educational features: - Drag and drop mode: Allow students to rearrange unit squares from a² and b² squares into the c² square - Show different visual proof styles (traditional, dissection, algebraic) - Include famous Pythagorean triples as preset buttons (3-4-5, 5-12-13, 8-15-17)
Default parameters: - Start with 3-4-5 triangle (most recognizable) - Grid visible by default - Animation speed: medium
Implementation: p5.js with animation using frameCount Include smooth transitions for the rearrangement animation
Pythagorean Triples
A Pythagorean triple is a set of three positive integers \(a\), \(b\), and \(c\) that satisfy the Pythagorean theorem: \(a^2 + b^2 = c^2\).
Common Pythagorean triples include:
- 3, 4, 5 (the most famous!)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
You can multiply any Pythagorean triple by the same number to get another triple. For example, multiplying 3-4-5 by 2 gives 6-8-10, which is also a Pythagorean triple.
Converse of the Pythagorean Theorem
The converse tells us how to determine if a triangle is a right triangle when we know all three side lengths.
If \(a^2 + b^2 = c^2\) (where \(c\) is the longest side), then the triangle is a right triangle.
If \(a^2 + b^2 > c^2\), then the triangle is an acute triangle (all angles less than 90°).
If \(a^2 + b^2 < c^2\), then the triangle is an obtuse triangle (one angle greater than 90°).
Triangle Type Classifier Table
| Side lengths | Calculation | Result | Triangle type |
|---|---|---|---|
| 3, 4, 5 | 3² + 4² = 9 + 16 = 25; 5² = 25 | 25 = 25 | Right |
| 6, 7, 8 | 6² + 7² = 36 + 49 = 85; 8² = 64 | 85 > 64 | Acute |
| 5, 6, 10 | 5² + 6² = 25 + 36 = 61; 10² = 100 | 61 < 100 | Obtuse |
Distance in the Coordinate Plane
The Pythagorean theorem gives us a powerful tool for finding distances on the coordinate plane!
To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we can imagine a right triangle where:
- One leg has length \(|x_2 - x_1|\) (horizontal distance)
- The other leg has length \(|y_2 - y_1|\) (vertical distance)
- The hypotenuse is the distance we want to find
Distance Formula
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
where:
- \(d\) is the distance between the two points
- \((x_1, y_1)\) is the first point
- \((x_2, y_2)\) is the second point
Example: Find the distance between \((2, 3)\) and \((5, 7)\).
\(d = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)