Chapter 3: The Pythagorean Theorem: All the Right Angles
by Dawn Griffiths, Stray
Head First 2D Geometry
- Advance Praise for Head First 2D Geometry
- Praise for other Head First books
- Copyright
- Dedication
- The Authors
- Table of Contents (1/2)
- Table of Contents (2/2)
- How to Use this Book: Intro
- Chapter 1: Finding Missing Angles: Reading Between the Lines
- There’s been a homicide
- In the ballistics lab you’ve got to cover all the angles
- Do the angles between Benny, Micky, and the bullet match up?
- Right angles aren’t always marked with numbers
- Angles can be made up of other, smaller angles
- Complementary angles always add up to a right angle (90º)
- Right angles often come in pairs
- Angles on a straight line add up to 180º
- Pairs of angles that add up to 180º are called supplementary angles
- Vertical angles are always equal
- The corner angles of a triangle always add up to a straight line
- Find one more angle to crack the case
- Something doesn’t add up!
- If it doesn’t all add up, then something isn’t as it seems
- You’ve proved that Benny couldn’t have shot Micky!
- We’ve got a new sketch—now for a new ballistics report
- We need a new theory
- Work out what you need to know
- Tick marks indicate equal angles
- Use what you know to find what you don’t know
- The angles of a four-sided shape add up to 360º
- Parallel lines are lines at exactly the same angle
- Parallel lines often come with helpful angle shortcuts
- Great work—you cracked the case!
- Your Geometry Toolbox
- Chapter 2: Similarity and Congruence: Shrink to Fit
- Welcome to myPod! You’re hired
- Liz wants you to etch her phone
- The designer noted some of the details
- The design tells us that some triangles are repeated
- Similar triangles don’t just look the same
- To use similarity, you need to be able to spot it
- You can spot similar triangles based on just two angles
- Employee of the month already?
- You sketch it—we’ll etch it!
- Fire up the etcher!
- The boss isn’t happy, but at least you’re not fired…
- It’s a problem of scale…
- Complex shapes can be similar, too
- You sketch it—we’ll etch it (to fit)
- Liz is back with a special request
- Similar shapes that are the same size are congruent
- Use what you know to find what you don’t know (1/2)
- Use what you know to find what you don’t know (2/2)
- Ratios can be more useful than sizes
- Ratios need to be consistent
- Your new design ROCKS!
- Your Geometry Toolbox
- Chapter 3: The Pythagorean Theorem: All the Right Angles
- Giant construction-kit skate ramps
- Standard-sized-quick-assembly-what?!?
- The ramps must have perpendicular uprights
- You can use accurate construction to test ramp designs on paper (1/2)
- You can use accurate construction to test ramp designs on paper (2/2)
- Not all lengths make a right triangle
- You can explore a geometry problem in different ways
- In geometry, the rules are the rules
- Any good jump has some similar scaled cousins
- The lengths of the sides are linked by a pattern
- The square of the longest side is equal to the squares of the other two sides added together
- The Pythagorean Theorem: a² + b² = c² (1/2)
- The Pythagorean Theorem: a² + b² = c² (2/2)
- Using Kwik-klik skate ramps is definitely the right angle!
- A longer rope swings further and lower
- So, how far can you swing on a six-meter rope?
- Your rope swing is perfect
- Your Geometry Toolbox
- Chapter 4: Triangle Properties: Between a Rock Show and a Triangular Place
- Everybody loves organizing a rock festival
- First we need to pick a venue
- Fencing costs money
- Does a bigger perimeter mean a bigger area?
- How many people can each venue hold?
- A triangle fits inside a bounding rectangle (1/2)
- A triangle fits inside a bounding rectangle (2/2)
- The area of a triangle = 1/2 base × height
- You’ve got $11,250 to spend
- All speakers are not created equal
- So what are you looking for in your speakers?
- The ideal speakers are wider and longer than the venue…but only by a little
- 100m will do, but can you rent the 60° speaker?
- The 60° speakers are spot on
- All that’s left is to pick a spot for the drinks stall
- A triangle has more than one center
- The center of a triangle can be outside the triangle
- Let’s put the drink stall at the centroid
- The rock festival is ready!
- The people behind the drinks stall won’t see the stage…
- You need a screen for less than $1,440
- Will the special offer screen still do the job?
- You can find area from sides using Hero’s formula
- Hero’s formula and “1/2 base × height” work together
- The rock festival is gonna…rock!
- Your Geometry Toolbox
- Chapter 5: Circles: Going Round and Round
- It’s not just pizza—it’s war!
- How does MegaSlice’s deal measure up?
- The diameter of a circle is twice its radius
- How do slices compare to whole pizzas?
- Sectors of a circle have angles totaling 360°
- MegaSlice’s $10 deal is a con!
- Pepperoni crust pizza—but at what price?
- The pepperoni perimeter is 3 (and a bit) times diameter
- Mario wants to put your pepperoni crust pricing formula to the test
- The customers are always fussy
- An arc is a section of the circumference
- Mario’s business is booming!
- But MegaSlice is at it again...
- We need to find the area of the two pizza deals
- Each sector (slice) is a triangle (kind of)
- Area of a circle = πr²
- Mario’s pizza is here to stay
- Your Geometry Toolbox
- Chapter 6: Quadrilaterals: It's Hip to be Square
- Edward’s Lawn Service needs your help
- Your first lawn
- The lawn is a parallelogram
- Let’s split the parallelogram
- Business is booming!
- If you don’t like what you’re given, change it
- But people are upset with Ed’s prices…
- Let’s compare the two lawns
- The lawns need edging, too
- Same shape, different perimeters
- Edward changed his rates…
- …and the customers keep flooding in
- Use diagonals to find the area of the kite
- Landowners, unite
- There are some familiar things about this shape
- Calculate trapezoid area using base length and height
- The quadrilateral family tree
- You’ve entered the big league
- Your Geometry Toolbox
- Chapter 7: Regular Polygons: It's All Shaping Up
- We need to choose a hot tub
- All the hot tubs are regular polygons
- Regular polygons have equal sides and angles
- Butt-space is all about perimeter
- Is 3 cubic meters of water a lot or a little?
- Hot tub volume is area × depth
- The hot tub’s area must be 6m²
- Which hot tub shape gives the most butt-space?
- Work backward from area to find butt-space
- Is 19.6 butts a lot or a little?
- The square tub beats the circle tub
- Two tubs down, five to go
- You’ve found the formula for the area of an equilateral triangle
- Keep track of complex comparisons with a table (1/2)
- Keep track of complex comparisons with a table (2/2)
- Chop the polygons into triangles
- What do we need to know about the polygon triangles?
- The circles give us the properties we need
- Polygon area = 1/2 perimeter × apothem
- More sides = fewer butts
- Rock stars—high maintenance?
- Great tub choice!
- But what about dimensions?
- It’s time to relax in the hot tub!
- Your Geometry Toolbox
- Leaving town…
- It’s been great having you here in Geometryville!
- Index (1/2)
- Index (2/2)
3
the pythagorean theorem
All the right angles
You just straighten one leg, then
straighten the other leg, clap your hands,
and you’re dancing the hypotenuse!
It’s all in the hips, really.
Sometimes, you really need to get things straight.
Ever tried to eat at a wobbly table? Annoying, isn’t it? There is an alternative to shoving
screwed-up paper under the table leg though: use the Pythagorean Theorem to make
sure your designs are dead straight and not just quite straight. Once you know this
pattern, you’ll be able to spot and create right angles that are perfect every time.
Nobody likes to pick spagetti out of their lap, and with the Pythagorean Theorem, you
don’t have to.
this is a new chapter 103
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