Thursday, January 29, 2015

2015 Mathcounts Chapter / State Prep : Octagon





Question #1: How to find the area of a regular octagon.
 method I:
The area of a regular octagon can be obtained by using 45-45-90 degree angle ratio to get the sides
and from there get the area.
method II: 
You can also use the area of the square minus the 4 triangles at the corner.
To get the length of the unknown
side length, again, use 45-45-90 degree angle ratio.

If the side is "a" unit, the area is \(2(1+\sqrt {2})*a^{2}\).



Question #2:  What is the area ratio of the regular octagon ABCDEFGH 
to the embedded square ACEG?

\(\Delta ATC\) is a right triangle and \(\overline {AC}\) is the hypotenuse.

Use the Pythagorean theorem:\(\left( \overline {AC}\right) ^{2}\) [same as the area of the square]
= \(\left( \sqrt {2}\right) ^{2}+\left( 2+\sqrt {2}\right) ^{2}\)
=\(8+4\sqrt {2}\) 
The aera ratio of the regular hexagon ABCDEFGH to the area of the embedded square ACEG
=\(\dfrac {8+8\sqrt {2}} {8+4\sqrt {2}}\) = \(\sqrt {2}\) 





Questions (answer key below)
#1: What is the area of a regular octagon if each side is "2"
#2: What is the area of the rectangle HCDG? 
#3: What is the area of the trapezoid ABCH?
#4: What is the area of the triangle ADG?










Answer key:
#1:  \(8+8\sqrt {2}\)
#2: Area of the rectangle HCDG = \(4+4\sqrt {2}\) [ 1/2 of the area of the octagon]
#3: Area of the trapezoid ABCH is \(2+2\sqrt {2}\) [ 1/4 of the area of the octagon]
Notice the area of the HCDG = the sum of the other two congruent trapezoids
ABCH and FEDG, thus break the octagon to 1 : 1 ratio.
#4: The area is \(4+3\sqrt {2}\)
 




Monday, January 26, 2015

2015 Mathcounts State/National Prep

Harder concepts from Mathcounts Mini :
Also try the follow-up problems with detailed solutions. Don't do every question. Just the ones you think is hard.

Geometry :
#27 : Area and Volume 

#30: Similar Triangles and Proportional Reasoning

#34: Circles and Right Triangles

#35: Using Similarity to Solve Geometry Problems

#41: Analytic Geometry : Center of Rotation

More questions to practice from my blog post. 

From Mathcounts Mini : Video tutorials on counting and probability for Mathcounts state/national prep concepts are in the of difficulty.

Counting the Number of Subsets of a Set

Constructive Counting

More Constructive Counting 

Probability and Counting

Probability with Geometry Representations  : Oh dear, the second half part is hilarious.

Probability with Geometry Representations : solution to the second half problem from previous video

Try this one from 1998 AIME #9. It's not too bad.

From Mathcounts Mink : Center of rotation, equal distance from a point to other two points or two lines, angle bisector (last problem on the follow-up worksheet)

Mathcounts Mini : #41 - Analytic Geometry