2D Applications of Pythagoras- Part 1
Slideshow 41, MathematicsMr. Richard Sasaki, Room 307
ObjectivesObjectives
• Review polygon properties and interior angles
• Review circle properties• Be able to apply the Pythagorean Theorem
to other polygons, circles and graphs
Pythagorean TheoremPythagorean Theorem
We know how to find missing edges of triangles now. Let’s apply this to other polygons. Oh wait, what is a polygon?A polygon is a 2D shape with straight edges only.We actually already did this, this is a review.ExampleFind the unknown value on the polygon below.
7𝑐𝑚7𝑐𝑚
3𝑐𝑚
𝑥𝑐𝑚𝑎2+𝑏2=𝑐232+72=𝑐2
9+49=𝑐258=𝑐2
𝑐=√58𝑥=√58
Answers (Question 1)Answers (Question 1)
60o
3𝑐𝑚
30o𝑥𝑐𝑚
𝑦 𝑐𝑚
First calculate , and then after.𝑎2+𝑏2=𝑐2⇒¿¿
32
94+𝑥2=9⇒ 𝑥2=
364−94
𝑥=√ 274¿ √27√4¿ 3√3
2
3√32
𝑐𝑚
𝑎2+𝑏2=𝑐2⇒( 3√32 )2
+𝑏2=(¿¿)2
3√3⇒ 274 +𝑏2=27
⇒𝑏2=1084−274⇒𝑏=√ 814¿ 92
𝑦=¿32+92=¿6𝑐𝑚
12
Answers (Question 2)Answers (Question 2)
𝑥
1 We know an interior angles in a hexagon add up to .720𝑜
() 720𝑜
6=¿120o
120o 30oLet’s calculate first.
1
6 0o
As we have a 30-60-90 triangle, .√32
So .
Answers (Question 3)Answers (Question 3)
𝑥
1
We know an interior angles in an octagon add up to .1080𝑜
() 1080𝑜
8=¿135o
45o
135o
1
1
𝑎
𝑎
Let’s calculate .We have a 45-45-90 triangle.
45𝑜
45𝑜1
√21
45𝑜
45𝑜
1
√211√2
𝑎=1
√2=¿√22
𝑥=¿√22
+1+ √22
¿1+√2
CirclesCirclesLet’s review some parts of the circle.
Centre (origin)
Radius
Tangent
Diameter
Chord
Question types with circles are simple, but it’s important that we understand the vocabulary.
CirclesCirclesExampleConsider a circle with radius 6 cm. The circle has a chord 8 cm long. Find the distance between the centre and the chord.
6𝑐𝑚
8𝑐𝑚
Note: It doesn’t matter where the chord is, as it is a fixed length, it is always the same distance from the centre. So it may as well touch the radius.
𝑥𝑐𝑚
𝑎2+𝑏2=𝑐2
𝑥2+42=62
𝑥2+16=36𝑥2=20𝑥=2√5
AnswersAnswers
𝑥=√112
𝑥=2√6
Radii touch tangents (or segments of tangents) at .
or
The length required is the shortest distance from the chord, so must be at .
2√5𝑐𝑚
GraphsGraphsDistances between points can be calculated, based on their coordinates. Note: The shortest distance between two points is always the of a triangle.hypotenuseExampleFind the distance between two points, and .
We should visualize the triangle like…or…
32
How did we get and ?3=¿5−22=¿7−5
Let’s find the hypotenuse.𝑎2+𝑏2=𝑐2⇒22+32=𝑐2
⇒13=𝑐2⇒𝑐=√13
Answers - EasyAnswers - Easy3√5 4√2√1702 √65
√2052 5
Answers - MediumAnswers - Medium
√ 41 √218 √2053√26 2√226 6 √1716√5 √794 2√557Distance is the greatest.
Answers - HardAnswers - Hard1. 2. We don’t know where in relation to the fountain that they are, 3. 4. We square the brackets,
FormulaeFormulaeSo simply for any and …
𝐴𝐵=√ (𝑥2− 𝑥1 )2+(𝑦 2− 𝑦1)2
Or written for the change in , and the change in , …
𝐴𝐵=√ (∆𝑥 )2+¿¿One of these will be provided in the exam, however you know how to do the questions without them anyway!Please bring a ruler, pencil and compass to the next lesson.