Subscribe

Mathematics Magazine for Grades 1-12  

 

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 half-hour 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, y-intercept, and t-intercept, 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)