## 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
=$$\dfrac {8+8\sqrt {2}} {8+4\sqrt {2}}$$ = $$\sqrt {2}$$

#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?

#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