Line Intercept, Slope, Area/Geometry Questions

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This question is similar to 2010 #3 Mathcouts Chapter Team problem. 

Solution I: Using the given slope, you can find the equation for AB ,
which is: y = -2 x + b, plug in (4, 7) and you have 7 = -2 x 4 + b so b = 15
The y intercept for AB is (0, 15)
Now you have four points, (0, 0), (0, 15), (4, 7) and (15, 0).
Using shoestring method to find area of any polygon, you have the area of polygon
AEDO = 82.5 square units.

Solution II:

After finding the y intercept for AB.

You break the area into three parts.

The side of the shaded region is 4 x 7 = 28.

The area of triangle ACE = 4 * (15-7) / 2 = 16

The area of triangle
EBD = (external height)7 * (15-4) / 2 = 38.5

So the total area is 82.5 square units.




Applicable problems: Answer key below.

#1:  The slope of AB is -3 and it intercepts with CD at point E, which is (3, 3). If the x intercepts of CD is (10,0), what is the area of AEDO?

#2: The slope of AB is 2 and it intercepts with CD at point E, which is (-2, 2 ). If the x intercepts of CD is (-7,0), what is the area of AEDO?

#3:  The slope of AB is -3 and it intercepts with CD at point E, which is ( 5, 8 ). If the x intercepts of CD is (14,0), what is the area of AEDO? -- Andrew's question.

#4: The slope of AB is \(\dfrac {-3} {2}\)and it intercepts with CD at point E, which is ( 2, 7 ). If the x intercepts of CD is (16,0), what is the area of AEDO? --Daniel's question.

#5: The slope of AB is -4/5 and it intercepts with CD at point E, which is (5, 2 ). If the x intercepts of CD is (12,0), what is the area of AEDO?












Answer key: 
#1: 33
#2: 13
#3: 113.5
#4: 66
#5: 27

Trianges That Share the Same Vertex/Similar Triangles

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Besides similar triangles, "triangles that share the same vertex" also appears regularly on geometry questions. 

Look , for example, at the left image. 
It's easy to see AD being the height to base CE for Δ AEC.

However, it's much harder to see the same AD being the external height of  Δ ABC if BC is the base.





Question to ponder based on the above image:
#1: If  BC : CE = 2 to 5 and the area of Δ ABE is 98, what is the area of Δ ABC and Δ ACE?

Solution:
Since both triangles share the same height AD, if you use BC and CE  as the bases, the area ratio stays constant as 2 to 5.  Thus the area of Δ ABC = (2/7) * 98 = 28 and the area of
Δ AEC = (5/7) *98 = 70.

Knowing the above concept would help you solve the ostensibly hard trapezoid question.

Question #2: If ABCD is a trapezoid where AB is parallel to CD,  AB = 12 units and CD = 18 units.
If the area of  Δ AEB = 60 square units, what is the area of ΔCED, Δ AED and ΔBEC?






Solution:  Δ AEB and ΔCED are similar (Why? Make sure you understand this?) Thus if AB to CD = 12 to 18 or 2 to 3 ratio (given), the ratio of the area of the two similar triangles is 2² to 3² = 4 to 9 ratio.
The area of  Δ AEB = 60 square units (given), so the area of ΔCED is 9 * (60/4) = 135 square units. 
Δ AEB and Δ AED share the same vertex. The height is the same with base BE and ED.
Thus the area ratio is still 2 to 3. The area of Δ AEB = 60 square units (given),
 the area of Δ AED = 3 * (60/2) = 90 square units.

Using the same reasoning you get the area of ΔBEC = 90 square units. [Keep in mind that Δ AED and ΔBEC don't have to be congruent, but they do have the same area.]

Similar Triangles I

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There are numerous questions on similar triangles at competition math.
Practice how to spot them and use them to solve tougher chapter, median hard state questions.

For example, There are three similar triangles in this image.

Triangle ABC is similar to triangle BDC and triangle ADB.

Using consistent symbols will help you set up the right proportion comparisons much easier.

BD² = AD x DC
BA² = AD x AC
BC² = CD x AC  

All because of the similarities.
That is also where those ratios work.











#1: What is the height to the hypotenuse of a 3-4-5 right triangle? 
Solution: 
 The area of a triangle is base x height over 2.
 Area of a 3-4-5 right triangle  = (3 x 4) / 2 = (5 x height to the hypotenuse) / 2
 Both sides times 2 : (3 x 4) = 5 x height to the hypotenuse
 The height to the hypotenuse = 12/5 

The same goes with 6-8-10 right triangle, the height to the hypotenuse is 4.8.
It's very straightforward so make sure you know how and why that works.

Hope this helps.

2007 State Harder Algebra Questions

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#16 2007 Mathcounts State Sprint: A data set for a class of 25 sixth graders has their ages listed as
the integer values of either 10 or 11 years. The median age in the data set is 0.36 years greater than the mean. How many 10-year-olds are in the class?

#23: 2007 Mathcounts State Sprint:  A box contains some green marbles and exactly four red marbles.
The probability of selecting a red marble is x%. If the number of green marbles is doubled, the probability of selecting one of the four red marbles from the box is (x - 15)%. How many green marbles are in the box before the number of green marbles is doubled?












#16: Solution I: 
 Since there are 25 students, the median has to be either 10 or 11 (Why?)
Since the mean has to be 10 (if all the children are all 10 year old) or higher, the median has to be 11.
Let there be x 10 year old students, and there are (25 - x) 11 year old students.

both sides * ----

 

 

Solution II: 

Let there be x 10 year old and y 11 year old. 

According to the given: 

10x + 11 y = 266
For x to be integer, y has to either 6 or 16. Further checking, only when y = 16, x = 9 works. 

#23: Let there be g numbers of Green marbles.
According to the given, we can set up two equations:

                                          equation 1

           equation 2

Equation 1 minus equation 2 and you have: 0 = xg + 60 -2xg + 30g
Move all the variables to the other side:
xg - 30g = g (x - 30) = 60
If g = 4, x = 45 (doesn't work)
If g = 6, x = 40 (it works.)

Mental Math Practices on Ratio, Percents and Proportion

Mental Math Practices: All questions can be solved in your head reasonably fast (less than 5 minutes) if you understand the concepts well.

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1. What is 12.5% of 128?

2. After 20% discount, you paid $144? What is the original price?

3. To make 20% profit, you charge your customer 144 dollars. What is the wholesale price?

4. You lost 20% in the stock market. What percentage increase will you break even?

5. A : B = 2 : 5 and A is 75 more than B, what is B?

6. A : B = 2 : 3 and A + B = 14, what is A and what is B?

7. A : B = 3 : 5 and B : C = 2 to 9, what is A : B?

8. Two triangles are similar and the length of their base ratio is 1: 2. The smaller triangle has a base that equals 5 and an area that is 110. What is the area of the larger triangle?

9. One side of the square increases 10% and the other side decreases 20 percent. What is the percentage change? Increase or decrease? By how much?

10. There are 60 students in a gym and 30% are boys. After a few more boys enter the gym, now 40% are boys. How many boys enter the room?