Download 16. ratios, proportions, and triangle inequality theorem

16. RATIOS, PROPORTIONS, AND
TRIANGLE INEQUALITY THEOREM
Ratios
When you are given a ratio such as 2 : 3 : 5, put an x after each number ( 2x, 3x, and 5x )
and set up an equation.
Examples
! of girls to the number of!boys is 3: 5. If there are
1) In a certain Math A class, the ratio of the number
a total of 32 students in this class, how many are girls and how many are boys?
!
!
The number of girls is 3x and the number of boys is 5x. Since the total number of students (girls + boys)
is 32, set up the equation :
3x + 5x = 32
8x = 32
8
8
x=4
Substitute to find the answer : Number of Girls = 3x = 3(4) = 12
Number of Boys = 5x = 5(4) = 20
!
2) The angles in a triangle are in the ratio of 1: 3 : 5. Find the number of degrees in the largest angle
!
in the triangle.
The three angles are 1x, 3x, and 5x. Since the angles of a triangle add up to 180º, set up the equation :
1x + 3x + 5x = 180
9x = 180
9
9
x = 20
!
!
Since 5x is the largest angle, substitute to find the answer : 5x = 5(20º ) = 100º
!
Proportions
!
Match up like parts (ex.
miles
hours
=
miles
hours
).
Examples
1) If four compact discs cost $27, at the same rate what is the cost of seven compact discs?
!
CDs
4 7
dollars
!
=
27 x
4 x = 189
4
4
x = 47.25
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$47.25
Created by J. Shahom (March 2005)
!
2) At a certain time during the day, light falls so that a pole 10 feet in height casts a shadow 15 feet
in length on level ground. At the same time, a man casts a shadow that is 9 feet in length. How tall
is the man?
height
shadow
!
10 x
=
15 9
90 = 15x
15 15
x =6
The man is 6 feet tall.
Triangle Inequality Theorem
!
In any triangle, the sum of any two sides must be more than the third side.
If you know all three sides, add any two of the sides and make sure that it is bigger than
the third side.
If you know two sides, add the two sides and subtract the two sides; the third side must be
between those two numbers.
Examples
1) Which set of numbers could represent the lengths of the sides of a triangle?
(1) {3, 6, 3}
(2) {3, 9, 14}
(1) {3, 6, 3}
!
!
3+ 3 = 6
(2) {3, 9, 14}
3 + 9 = 12
6 > 6 no
(3) {3, 5, 7}
(3) {3, 5, 7}
3+ 5 = 8
3 + 7 = 10
5 + 7 = 12
(4) {1, 2, 3}
sum of first two sides > third side
8 > 7 yes
10 > 5 yes
12 > 3 yes
sum of first and last sides > second side
sum of last two sides > first side
12 > 14 no
!
!
(4) {1, 2, 3}
1+ 2 = 3
3 > 3 no
In any triangle, all three of these must be true.
If even one of them is not true [as in choices
(1), (2), and (4)], then the numbers cannot
represent the sides of a triangle.
2) If the lengths of two sides of a triangle are 4 and 8, the length of the third side cannot be
(1) 5
(2)!6
(3) 7
(4) 4
Add the two sides : 4 + 8 = 12
!
!
Subtract the two two sides : 8 " 4 = 4
So, the third side must be greater than 4 and less than 12.
The length of the third side could be 5, 6, or 7 since each of these numbers is greater than 4 and less than 12.
Since 4 is not greater than 4, the length of the third side cannot be 4.
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Created by J. Shahom (March 2005)