The formula for the midsegment of a trapezoid is midsegment = 1/2(base1 + base2). Find the value of x. 3x – 1 145 19 14.5 = √(3x + 1 + 19) Solve for x. Back Next Type here to search
The Correct Answer and Explanation is:
To solve for xxx, we use the formula for the midsegment of a trapezoid:Midsegment=12(base1+base2)\text{Midsegment} = \frac{1}{2} (\text{base}_1 + \text{base}_2)Midsegment=21(base1+base2)
From the problem:
- One base is 3x−13x – 13x−1
- The other base is 191919
- The midsegment is 14.514.514.5
Step 1: Plug into the formula
14.5=12((3x−1)+19)14.5 = \frac{1}{2} \left( (3x – 1) + 19 \right)14.5=21((3x−1)+19)
Step 2: Simplify the expression inside the parentheses
14.5=12(3x+18)14.5 = \frac{1}{2} (3x + 18)14.5=21(3x+18)
Step 3: Eliminate the fraction by multiplying both sides by 2
2⋅14.5=3x+182 \cdot 14.5 = 3x + 182⋅14.5=3x+1829=3x+1829 = 3x + 1829=3x+18
Step 4: Solve for xxx
29−18=3×29 – 18 = 3×29−18=3×11=3×11 = 3×11=3xx=113x = \frac{11}{3}x=311
Final Answer:
x=113\boxed{x = \frac{11}{3}}x=311
Explanation (like in a textbook, ~300 words):
In a trapezoid, the midsegment (also called the median) is the line segment that connects the midpoints of the two non-parallel sides. This segment is always parallel to the two bases and its length is the average of the lengths of the two bases. The formula to calculate the length of the midsegment is:Midsegment=12(base1+base2)\text{Midsegment} = \frac{1}{2} (\text{base}_1 + \text{base}_2)Midsegment=21(base1+base2)
In this problem, you are given the lengths of the two bases and the midsegment, with one base expressed in terms of the variable xxx. Specifically, base 1 is 3x−13x – 13x−1, base 2 is 19, and the midsegment is 14.5.
To find the value of xxx, substitute these values into the midsegment formula:14.5=12((3x−1)+19)14.5 = \frac{1}{2} \left((3x – 1) + 19\right)14.5=21((3x−1)+19)
Combine like terms inside the parentheses:14.5=12(3x+18)14.5 = \frac{1}{2} (3x + 18)14.5=21(3x+18)
Then, multiply both sides by 2 to remove the fraction:29=3x+1829 = 3x + 1829=3x+18
Isolate the variable by subtracting 18 from both sides:11=3×11 = 3×11=3x
Finally, divide both sides by 3:x=113x = \frac{11}{3}x=311
This is the value of xxx that makes the length of the midsegment equal to 14.5, based on the formula derived from trapezoid geometry.
