Harder Geometry Question from National Mathcounts

Question: Similar to a 96 National Mathcounts question: Circles of the same colors are congruent and tangent to each other. What is the ratio of the area of the largest circle to the area of the smallest circle?

Solution :
The most difficult part might be to find the leg that is "2R - r".
R and r being the radius of the median and the smallest circle.

Using Pythagorean theorem, you have \(\left( R+r\right) ^{2}= R^{2}+\left( 2R-r\right) ^{2}\)
\(R^{2}+2Rr+r^{2}=R^{2}+4R^{2}-4Rr+r^{2}\)
Consolidate and you have 6Rr = \(4R^{2}\)\(\rightarrow\) 3r = 2R \(\rightarrow\) r =\(\dfrac {2R} {3}\) The ratio of the area of the largest circle to the area of the smallest circle is thus \(\rightarrow\)\(\dfrac {\left( 2R\right) ^{2}} {\left( \dfrac {2R} {3}\right) ^{2}}\) = 9 .

2013 Mathcounts State Prep: Counting and Probability

Question #1: Rolling two dice, what is the probability that the product is a multiple of 3?
Solution I :
As long as one of the two numbers turn up as 3 or multiple of 3 (in this case, a "6"), the product of the two numbers will be a multiple of 3.
There are 6 * 6 = 36 ways to get 2 numbers. Out of the 36 pairs, you can have
3 - 1
3 - 2
3 - 3
3 - 4
3 - 5
3 - 6, 6 ways.
However, there are only 5 ways left if the other die has a 3 since 3 - 3 only counts as once.
1 - 3
2 - 3
4 - 3
5 - 3
6 - 3
Next we look at "at least one number is "6".
6 - 1
6 - 2
6 - 4 (We already used 6 - 3)
6 - 5
6 - 6 so 5 ways.
The other way around, we have
1 - 6
2 - 6
4 - 6
5 - 6  Total 4 ways, so the answer is \(\frac{\Large{( 6 + 5 + 5 + 4 )}}{\Large{36}}\) = \(\frac{\Large{ 20}}{\Large{36}}\) = \(\frac{\Large{5}}{\Large{9}}\).

Solution II: 
The easiest way to solve this problem is to use complementary counting, which is 1 (100% or total possible way) - none of the the multiples of 3 showing up, so 1 - \(\frac{\Large{4}}{\Large{6}}\) *\(\frac{\Large{4}}{\Large{6}}\) = 1 - \(\frac{\Large{2}}{\Large{3}}\) * \(\frac{\Large{2}}{\Large{3}}\)=  \(\frac{\Large{5}}{\Large{9}}\)

Question #2: [2002 AMC-12B #16] Juan rolls a fair regular eight-sided die. Then Amal rolls a fair regular six-sided die. What isthe probability that the product of the two rolls is a multiple of 3?
Solution:
Using complementary counting (see solution II of the previous question), 1 - \(\frac{\Large{6}}{\Large{8}}\) *\(\frac{\Large{4}}{\Large{6}}\) = 1 - \(\frac{\Large{3}}{\Large{4}}\) * \(\frac{\Large{2}}{\Large{3}}\)=  \(\frac{\Large{1}}{\Large{2}}\)

2013 Mathcounts State Prep: Similar Triangles and Height to the Hypotenuse

There are many concepts you can learn from this image, which cover numerous similar right triangles, ratio/proportion/dimensional change and the height to the hypotenuse.

Question:
#1: Δ ABC is a 3-4-5 right triangle. What is the height to the hypotenuse? 
Solution: 
Use the area of a triangle to get the height to the hypotenuse. 
Let the height to the hypotenuse be "h"
The area of Δ ABC is  \(\Large\frac{3*4}{2}\)= \(\Large\frac{5*h}{2}\)
Both sides times 2 and consolidate: h =  \(\Large\frac{3*4}{5}\) = \(\Large\frac{12}{5}\)

Practice: What is the height to the hypotenuse?

Question:
#2: How many similar triangles can you spot?
Solution: 
There are 4 and most students have difficulty comparing the largest one with the other smaller ones.
Δ ABC is similar to Δ ADE, Δ FBD, ΔGEC. Make sure you really understand this and can apply this to numerous similar triangle questions. 

Question: 
#3: What is the area of  □ DEGF if \(\overline{BF}\) = 9 and \(\overline{GC}\) = 4
Solution: 
Using the two similar triangles Δ FBD and  ΔGEC (I found using symbols to find the corresponding legs
 to be much easier than using the lines.), you have \(\frac{\Large{\overline{BF}}}{\Large{\overline{FD}}}\) = \(\frac{\Large{\overline{GE}}}{\Large{\overline{GC}}}\).
s (side length of the square) = \({\overline{GE}}\) =  \({\overline{FD}}\)
Plug in the given and you have 9 * 4 = s² so the area of □ DEGF is 36 square units. (each side then is square root of 36 or 6)

Question: 
#4: Δ ABC is a 9-12-15 right triangle. What is the side length of the square? 
Solution :
The height to the hypotenuse is\(\frac{\Large{9*12}}{\Large{15}}\) = \(\frac{\Large{36}}{\Large{5}}\)
Δ ABC is similar to Δ ADE. Using base and height similarities, you have \(\frac{\Large{\overline{BC}}}{\Large{\overline{DE}}}\) = \(\frac{\Large{15}}{\Large{S}}\) = \(\frac{\frac{\Large{36}}{\Large{5}}}{\frac{\Large{36}}{\Large{5}} - \Large{S}}\)
Cross multiply and you have 108 - 15*S = \(\frac{\Large{36}}{\Large{5}}\) *S
S =\(\frac{\Large{180}}{\Large{37}}\)

Ratio, Proportions -- Beginning Problem Soving (SAT level)

Questions:
#1: At a county fair, if adults to kids ratio is 2 to 3 and there are 250 people at the fair. How many adults and how many kids? 

Solution I :  Adults to Kids = 2 : 3 (given). Let there be 2x adults and 3x kids (only when two numbers have the same multiples can you simplify the two number to relatively prime.
 2x + 3x = 250;  5x = 250; x = 50 so there are 2x or 2 * 50 = 100 adults and 3x or 3 * 50 = 150 kids.

Solution II:  From the given information, you know that every time when there are 5 parts (2 + 3 = 5), there
will be 2 parts for A, or   of the total, and 3 parts for B, or of the total.
  * 250 = 100 adults and * 250 = 150 kids

#2: In a mixture of peanuts and cashews, the ratio by weight of peanuts to cashews is 5 to 2. How many pounds of cashews will there be in 4 pounds of this mixture? (an actual SAT question)

Solution:
You can use either the first or second method (see above two solutions) but the second solution is much faster. 
* 4 = pounds

#3: Continue with question #1: How many more kids than adults go to the fair? 

Solution I: 
Use the method on #1 and then find the difference.
150 - 100 = 50 more kids.
  
Solution II: 
Using the method for #1: solution II, you know    of the all the people go to the fair are adults ,
and of the total people go to the fair are kids.
(  -   ) * 250 = 50 more kids.

#3: If the girls to boys ratio at an elementary school is 3 to 4 and there are 123 girls, how many boys are there at that elementary school? 
Solution:
This one is easy, you set up the equation. Just make sure the numbers line up nicely.
Let there be "x" boys. = . Cross multiply to get x. Or since 123 is 41 * 3;
4 * 41 = 164 boys

#4 : If the girls to boys ratio at an elementary school is 2 to 5 and there are 78 more boys than girls, how many girls are there at the elementary school? 

Solution I: 
Again, you can use the algebra and let there be 2x girls and 5x boys. 
According to the given, 5x - 2x = 3x = 78 so x = 26
Plug in and you get there are 2*26 = 52 girls and 5*26 = 130 boys.

Solution II: 
If you keep expanding the ratio using the same multiples, 2 : 5 = 4 : 10 : 6 : 15...
Do you see that the difference of boys and girls are always multiples of 3 ( 5 - 2  = 3).
  = 26 so there are 2 * 26 or 52 girls

Review: Shoestring method, Heron's formula, Pythagorean Triples and Others related to Area of a triangle or polygon

Please give me feedback if there is any error or what concepts to be included for future practice tests. Thanks a lot in advance.

Online practices on finding the area of a triangle or irregular polygons.
Click here for the timed online test. Type in your first name and only write down "number" answers.

Before trying out the online timed test, here are some reminders.

a. If the irregular polygon include the origin, then using that as a starting point for shoestring method might save some calculation time.

b. There are Pythagorean Triples hidden in lots of questions that ask for the area of a triangle given three side lengths. Thus, if you find the side length that matches the triples or multiples of the triples, it might be much easier to use that instead of heron's formula.

c. For quadrilaterals, if the diagonals are perpendicular to each other, such as rhombus, kite or others, it's much easier to use D1* D2 / 2 to get the area.

d. Sometimes breaking the polygon into different smaller parts is the easiest way to find the answer.

e. Deliberate practices are the key to steady progress.

Have fun problem solving!!