Extended Ratios
Remember, ratios compare two quantities but sometimes more quantities than two quantities need to be compared.
For example: A triangle has three angles. IF you want to compare m∠D to m∠E to m∠F then you will have an extended ratio 70° to 60° to 50° or 70:60:50.
However, do not write as a fraction 70/60/50
Now you can REDUCE the extended ratio as long as you have a factor for all three.
These numbers have a common factor of 10.
Example
Problem: The measures of the angles of a triangle are in the extended ratio of 4:5:9. Find the measures of the angles.
This extended ratio was reduced or divided by a common factor. However, we don't know what that factor was. Let's represent it with an x. Now, we can represent the extended ratio as 4x:5x:9x.
Do you remember the sum of the angles in a triangle?
___°
Answer: 180
To solve the problem we can write an equation 4x + 5x + 9x = 180
___ = 180 Combine like terms
Answer: 18x
x = ___ Divide each side by 18 to isolate x.
Answer: 10
We've found x, which is the common factor, but we aren't done yet! The question asks us to find the measure of the angles, so we have to substitute the value of the common factor into our extended ratio.
Now each angle would be:
4x = 4( ___ ) = ___
Answer: 10, 40
5x = 5( ___ ) = ___
Answer: 10, 50
9x = 9( ___ ) = ___
Answer: 10, 90
Your Turn #4
Problem: The measures of the angles of a triangle are in the extended ratio of 2:5:8. Find the measures of the angles.
If we use 2x to represent the first angles, how can we represent the next two angles? ___ and ___
Answer: 5x, 8x
Ok, so now we can set up and solve the equation.
2x + 5x + ___ = ___
Answer: 8x, 180
Combine like terms
___ = 180
Answer: 15x
x = ___
Isolate x
Answer: 12
Have you answered the question now that you know x = 12?
Yes or No
Answer: No ; Knowing that x = 12 is not enough information to answer the question. 12 is simply the common factor in the extended ratio. The question asks us for the measure of the angles. So, substitute x back into our original expressions for the angles (2x, 5x, 8x) to answer the question.
Ok, let's substitute x into the original expressions for the angles to complete the problem.
2x = 2( ___ ) = ___
Answer: 12, 24
5x = 5( ___ ) = ___
Answer: 12, 60
8x = 8( ___ ) = ___
Answer: 12, 96
