If two sides of a triangle are 12 cm and 20 cm, the third side must be larger than ________ cm and smaller than ________ cm.
Fill in the blank and explain, please and thank you!
Property of any triangle:
The sum of any two sides of the triangle must be greater than the third side.
so
1. x+12>20 or x > 8
2. 12+20 > x or x < 32
3. x+20 > 12 or x > -8 , well that's obvious
so it looks like 8 < x < 32
check: let x=10
is 10+12 > 20 yes
is 10+20 > 12 yes
is 20+10 > 12 yes
check: let x = 5 (outside our domain)
is 5+10 > 20 NO!
check: let x = 40 (outside our domain on the other side)
is 10+12 > 40 NO.
To determine the possible range for the third side of a triangle, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have two sides of lengths 12 cm and 20 cm. So, the sum of these two sides must be greater than the length of the third side:
12 cm + 20 cm > third side
Simplifying the equation, we have:
32 cm > third side
Therefore, the third side must be larger than 32 cm.
On the other hand, since it is given that the side lengths of the triangle are 12 cm and 20 cm, the third side cannot be larger than the sum of these two side lengths:
third side < 12 cm + 20 cm
third side < 32 cm
Therefore, the third side must be smaller than 32 cm.
In conclusion, the third side of the triangle must be larger than 32 cm and smaller than 32 cm.
To find the range of possible values for the third side of a triangle, also known as the "triangle inequality," you can use the following rule:
The third side of a triangle must be larger than the difference between the lengths of the other two sides and smaller than the sum of their lengths.
In this case, we have two sides of lengths 12 cm and 20 cm. Therefore, the third side must be larger than the difference between these two sides (20 cm - 12 cm = 8 cm) and smaller than the sum of their lengths (20 cm + 12 cm = 32 cm).
So, the third side of the triangle must be larger than 8 cm and smaller than 32 cm.