DF = 3x + 1, DE = EFm = x +5.
The perimeter is 31.
Find x, DE and DF.
assuming we're talking abou a triangle DEF, just add up the sides and solve for x:
3x+1 + x+5 + x+5 = 31
5x + 11 = 31
x = 4
DE=EF=9
DF = 13
To find the values of x, DE, and DF, we can use the information given about the perimeter.
Let's begin by determining the lengths of DE and EF.
Given: DE = EF = x + 5 (since mEF = EF)
Let's assign a variable for the length of DE:
Let DE = a
Therefore, EF = a.
Now, let's determine the length of DF using the given equation:
DF = 3x + 1
To calculate the perimeter, we add the lengths of all three sides:
Perimeter = DE + DF + EF
From the given information, the perimeter is 31:
31 = a + (3x + 1) + a
Simplifying the equation:
31 = 2a + 3x + 1
Rearranging the terms:
30 = 2a + 3x
Now, we have two variables, a and x, in the equation. To solve for x and a, we need an additional equation. We can use the fact that DE = EF = x + 5 to derive one.
Since DE = x + 5, and DE was also assigned as a, we have:
a = x + 5
Now, we have two equations:
Equation 1: 30 = 2a + 3x
Equation 2: a = x + 5
We can substitute the value of a from Equation 2 into Equation 1:
30 = 2(x + 5) + 3x
Distributing:
30 = 2x + 10 + 3x
Combining like terms:
30 = 5x + 10
Subtracting 10 from both sides:
20 = 5x
Dividing both sides by 5:
x = 4
Now that we know the value of x, we can substitute it back into Equation 2 to find a:
a = x + 5
a = 4 + 5
a = 9
Therefore, x = 4, DE = 9, and DF = 3x + 1 = 3(4) + 1 = 13.