In the previous section we saw some solved examples on proportionality between two quantities. In this section we will see a different type of proportion.
1. Consider a regular polygon. We know how to calculate the sum of all it’s interior angles.
2. The formula is s = 180(n-2).
• Where s is the sum
• n is the number of sides of the regular polygon.
3. Let us use this formula for a triangle:
• For a triangle, n = 3
• So s = 180 × (3-2) = 180 × 1 = 180
4. Let us use the formula for a square:
• For a square, n = 4
• So s = 180 × (4-2) = 180 × 2 = 360
5. Let us use the formula for a pentagon:
• For a pentagon, n = 5
• So s = 180 × (5-2) = 180 × 3 = 540
6. Let us tabulate the results:
From the table we can see that s/n is not a constant. So s is not proportional to n.
7. Let us modify the formula a little:
Let s = 180 × m
• Where s is the sum
• m = (n-2)
• n is the number of sides of the regular polygon.
Now the tabulation will be as shown below:
We can see that s is proportional to m. The constant of proportionality is 180.
8. In ordinary language, we can say this:
The sum of interior angles of a regular polygon is proportional to ‘2 less than the number of sides’.
There are many examples where proportionality can be achieved by making modifications to one quantity. Let us see another example:
■ We have seen that the area of a square is not proportional to it’s side. Details here. The table that we saw is shown here again.
• We know that the area of a square is not proportional to it’s side because a/s ratio is not a constant.
• But if we put p = s2, a new table can be formed:
• We can see that a is proportional to p. The constant of proportionality is 1.
• In ordinary language, we can say this:
Area of a square is proportional to the square of it’s side.
We will now see some solved examples:
Solved example 24.8
For circles, is the area proportional to the square of the radius? If so, what is the constant of proportionality?
Solution:
1. We have seen that the area of a circle is not proportional to it’s radius. Details here.
But the area may be proportional to the 'square of the radius'. Let us try:
2. We know that area a = πr2
• Put r2 = q
• Then a = πq
♦ π is a constant
♦ when q increases a increases
♦ When q decreases a decreases
3. So a is proportional to q. That means, a is proportional to the square of the radius.
4. The constant of proportionality is π.
Solved example 24.9
For equilateral triangles, is the area proportional to the square of the side? If so, what is the constant of proportionality?
Solution:
1. We know that area of an equilateral triangle is given by a = (√3⁄4)s2 Details here.
Where s is the length of side.
• Put s2 = q
• Then a = (√3⁄4)q
♦ √3⁄4 is a constant
♦ when q increases a increases
♦ When q decreases a decreases
2. So a is proportional to q. That means, a is proportional to the square of the side.
3. The constant of proportionality is √3⁄4.
So we find that, in some cases, proportionality can be achieved by modifying one quantity. In the next section we will see Inverse proportions.
1. Consider a regular polygon. We know how to calculate the sum of all it’s interior angles.
2. The formula is s = 180(n-2).
• Where s is the sum
• n is the number of sides of the regular polygon.
3. Let us use this formula for a triangle:
• For a triangle, n = 3
• So s = 180 × (3-2) = 180 × 1 = 180
4. Let us use the formula for a square:
• For a square, n = 4
• So s = 180 × (4-2) = 180 × 2 = 360
5. Let us use the formula for a pentagon:
• For a pentagon, n = 5
• So s = 180 × (5-2) = 180 × 3 = 540
6. Let us tabulate the results:
From the table we can see that s/n is not a constant. So s is not proportional to n.
7. Let us modify the formula a little:
Let s = 180 × m
• Where s is the sum
• m = (n-2)
• n is the number of sides of the regular polygon.
Now the tabulation will be as shown below:
8. In ordinary language, we can say this:
The sum of interior angles of a regular polygon is proportional to ‘2 less than the number of sides’.
There are many examples where proportionality can be achieved by making modifications to one quantity. Let us see another example:
■ We have seen that the area of a square is not proportional to it’s side. Details here. The table that we saw is shown here again.
• But if we put p = s2, a new table can be formed:
• We can see that a is proportional to p. The constant of proportionality is 1.
• In ordinary language, we can say this:
Area of a square is proportional to the square of it’s side.
We will now see some solved examples:
Solved example 24.8
For circles, is the area proportional to the square of the radius? If so, what is the constant of proportionality?
Solution:
1. We have seen that the area of a circle is not proportional to it’s radius. Details here.
But the area may be proportional to the 'square of the radius'. Let us try:
2. We know that area a = πr2
• Put r2 = q
• Then a = πq
♦ π is a constant
♦ when q increases a increases
♦ When q decreases a decreases
3. So a is proportional to q. That means, a is proportional to the square of the radius.
4. The constant of proportionality is π.
Solved example 24.9
For equilateral triangles, is the area proportional to the square of the side? If so, what is the constant of proportionality?
Solution:
1. We know that area of an equilateral triangle is given by a = (√3⁄4)s2 Details here.
Where s is the length of side.
• Put s2 = q
• Then a = (√3⁄4)q
♦ √3⁄4 is a constant
♦ when q increases a increases
♦ When q decreases a decreases
2. So a is proportional to q. That means, a is proportional to the square of the side.
3. The constant of proportionality is √3⁄4.
No comments:
Post a Comment