Flashcards in Chapter 5: Area Deck (21):

1

## What are the postulates for polygonal areas?

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1. To each simple polygon is associated a non-negative number called its area

2. Invariance: Congruent polygons have the same area

3. The area of the union of a finite number of non-overlapping polygons is the sum of the areas of the individual polygons

4. Rectangular area: The area of an a*b rectangle is ab

2

## Thm 5.1.2:

### The area of an a*a square is a^2.

3

## Thm 5.1.3:

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The area of a parallelogram with bas b and altitude h is bg.

Pf provided.

4

## Thm 5.1.4:

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The area of a triangle with base b and altitude h is bh/2.

Pf provided.

5

## Thm 5.1.5:

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The area of a triangle with inradius and perimeter p is rp/2.

Pf provided.

6

## New pfs for old concepts...

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Ex 5.1.7: Pythagoras' Thm.

Thm 5.2.5: A triangle's 3 medians are concurrent.

Corollary 5.2.8: Pythagoras' Thm (Uses Ex 5.2.7).

7

## Thm 5.1.8:

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The area of a circle is half the product of its radius and its circumference.

A = Cr/2

Pf provided.

8

## Thm 5.2.1:

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Let ℓ || m, A, B, E, F ∈ ℓ, and C, D, G, H ∈ m s.t. ABCD and EFGH are parallelograms. Then [ABCD] / [EFGH] = AB/EF.

Pf provided.

9

## Corollaries 5.2.2-5.2.4 (Triangles of Equal Height):

### See Notes for Further Examples...

10

## Ex 5.2.6:

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Let D be a pt on BC. A line through D which bisects the area of △ABC is:

Case 1: D=M, DA bisects

Case 2: D between BM, DP bisects where P is pt on AC at intersection of line through M parallel to line DA.

Soln provided.

11

## Thm 5.2.9 (The Angle Bisector Thm):

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Let D ∈ BC and A not on BC, then

1. AB/AC = BD_line / DC_line iff AD bisects ∠BAC

2. AB/AC = -BD_line / DC_line iff AD bisects Exterior angle of A.

Pf provided.

12

## Thm 5.2.10:

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Let D be on BC and A a pt not on BC. If E is on AD, then

[ABE] / [ACE] = BD/CD

Pf provided.

13

## Recall, for right triangles:

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sin(x) = y/h (opp./hyp.)

cos(x) = x/h (adj./hyp.)

For 90≤x≤180, sin(x) = sin(180-x)

cos(x) = -cos(180-x)

14

## Thm 5.3.1 (Law of Sines):

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In △ABC,

a/sin(A) = b/sin(B) = c/sin(C) = 2R

where R is the circumradius.

Pf provided.

15

## Thm 5.3.2 (SAS case):

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The area of a triangle △ABC is [ABC] = (1/2) absin(C).

Pf provided.

16

## Ex 5.3.3:

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[ABC] = abc/4R, where R is the circumradius.

Soln provided.

17

## Thm 5.3.4 (ASA Case):

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The area of △ABC is [ABC] = a^2 sin(B) sin(C) / 2sin(A)

(can always get 3rd angle from other 2)

Pf provided.

18

## Thm 5.3.2 (SSA+ Case):

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Assuming a>b, the area of △ABC is

[ABC] = (1/2) b sin(A) [sqrt( a^2 - b^2 sin(A)^2) + bcos(A)]

Pf provided.

19

## Thm 5.3.6 (SSS Case - Heron's Formula):

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Let s denote the 'semi-perimeter' of △ABC (i.e. s= (a+b+c)/2) , then:

[ABC] = sqrt( s(s-a)(s-b)(s-c) ).

Pf provided.

20

## Thm 5.3.7 (The Law of Cosines):

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In △ABC, c^2 = a^2 + b^2 - 2ab cos(C).

Pf provided.

21