please help!!
A second ant walks all the way around a triangle drawn in the dirt. Two of the dimensions of the triangle are 5 1/10 centimeters and 7 1/5 centimeters. The total distance traveled by the ant is 20 centimeters. What is the length of the third side of the triangle?
Let's assume the third side of the triangle has a length of x centimeters.
The second ant walks all the way around the triangle, so it travels the distance of all three sides combined.
The first side has a length of 5 1/10 centimeters, which can also be written as 51/10 centimeters.
The second side has a length of 7 1/5 centimeters, which can also be written as 36/5 centimeters.
Therefore, the total distance traveled by the ant is:
51/10 + 36/5 + x
We know that the total distance traveled is 20 centimeters, so we can set up an equation:
51/10 + 36/5 + x = 20
To solve for x, we need to make sure the equation has a common denominator.
Multiplying both sides of the equation by 10, we get:
51 + 72 + 10x = 200
Combining like terms, we have:
123 + 10x = 200
Subtracting 123 from both sides, we get:
10x = 77
Dividing both sides by 10, we get:
x = 7.7
Therefore, the length of the third side of the triangle is 7.7 centimeters.
To solve this problem, we can use the perimeter of the triangle and subtract the lengths of the given sides to find the length of the third side.
1. Write down the lengths of the given sides:
- Side 1: 5 1/10 centimeters
- Side 2: 7 1/5 centimeters
2. Convert these lengths to improper fractions:
- Side 1: 5 1/10 = 51/10 centimeters
- Side 2: 7 1/5 = 36/5 centimeters
3. Add the lengths of the given sides to find the perimeter:
Perimeter = Side 1 + Side 2 + Side 3
Using the formula, we can write the equation as:
20 = 51/10 + 36/5 + Side 3
4. Simplify the equation:
To add the fractions, we need to have a common denominator. The least common multiple of 10 and 5 is 10.
20 = (51/10) + (72/10) + Side 3
20 = (51 + 72) / 10 + Side 3
20 = 123/10 + Side 3
5. Now, isolate Side 3 by subtracting 123/10 from both sides:
20 - 123/10 = 123/10 - 123/10 + Side 3
(200/10) - (123/10) = Side 3
(77/10) = Side 3
Therefore, the length of the third side of the triangle is 7 7/10 centimeters, or when converted to a mixed number, 7.7 centimeters.