Mathematics Magazine for Grades 112 



9/2004 



Grade 9
Theory:The bisector of an angle of a triangle divides the
opposite side into segments that are proportional to the adjacent sides.
That is, for any triangle ABC, the bisector of the angle at C divides the opposite side into segments of length x and y such that
Hint: Draw AE parallel to CD. Solutions from the Previous Issue:1. How to find the number of a given factorial? For example, the factorial of 4 is 24; (4 3 2 1), but how can we find a number if the factorial is given? Solution: Probably the easiest way is to find the prime factors of the number. For example, given 479,001,600 we could find the prime factors 10 5 2 1 1
479,001,600 = 2 3
5 7
11 Now, note that 11 appears, but 13 doesn't. So this narrows things down; it can only be 11! or 12!. That is, the largest prime in the prime factorization gives you a lower bound. The smallest missing prime gives us an upper bound. 2. The area of a circle is 16 pi. What is the circumference of the circle? 1) 8 pi 3) 16 pi 2) 2 pi 4) 4 pi Solution:The area of a circle and its circumference are both related to its radius. Here are the equations: circumference = pi 2 radius; area = pi radius radius. If the area of the circle is 16 pi, then we can use area = pi radius radius to find that the radius is 4. Next we can use the equation
circumference = pi
2
radius to find that: circumference = pi
2
4 = 8pi 3. The value of 5! Is: 1) 120; 2) 25; 3) 15; 4) –5 Solution: 5! =5
4
3
2
1=120 4. Miss Davis asked her students each to draw a picture describing their hobby. She asked them to use only three colors. Patrick had a box of crayons containing the colors red, blue, yellow, orange, green and purple. How many different ways can Patrick use his crayons to draw his picture? Solution: The answer is 20. The formula for calculating C(6,3) is = 20 This formula is derived as
follows: You can choose the first color in 6 ways from the 6 available. For the
second choice you now have 5 colors to choose from, so there are 5 ways to
choose the second color and finally 4 ways to choose the third color. So the top
line of our calculation is 6 x 5 x 4 = 120 We must now show why this answer is too large by a factor of 1 x 2 x 3. Suppose one of our choices was rby. Then other choices could be ybr, or byr or ryb and so on. In fact with 3 different colors there are 6 different ways of arranging them. (We could choose first position in 3 ways, second position in 2 ways, third position in 1 way giving 3 x 2 x 1 = 6 ways). But ALL 6 of these arrangements still count as ONE choice of 3 colors, and so the number 120 is too large by a factor of 6.It follows that the number of groups of 3 colors that can be chosen from 6 different colors is: = 20 Proposed
Exercises: 1.
Suppose that you are driving to Toronto at a constant speed, and
notice that after you have been traveling for 1 hour , you pass a sign saying
110 km to Toronto; and after driving another halfhour you pass a sign saying 85
km to Seattle. Using the horizontal axis for the time t and the vertical axis
for the distance y from Toronto, graph and find the equation y = mt + b for your
distance from Toronto. Find the slope, yintercept, and tintercept, and
describe the practical meaning of each. Find the distance between the two points 2. (1, 1) and (1, 1) 3.
(7, 4) and (10, 4) 4.
(8, 5) and (8, 0) 
