If the length of one base of a trapezoid is 5, the length of the other base is 2x + 7, and the length of the midsegment is 6x – 24, what is the value of x? Show equations and all work that leads to your answer.
The midsegment of a trapezoid is the arithmetic mean of the two parallel sides
http://www.9math.com/book/midsegment-trapezoid
so 6x-24 = (5 + 2x+7)/2
12x - 48 = 2x + 12
10x = 60
x = 6
To find the value of x, we can set up an equation by equating the length of the midsegment to the average of the lengths of the bases. Here's how:
First, let's define the midsegment of a trapezoid. A midsegment is a line segment that connects the midpoints of the two non-parallel sides of a trapezoid. In this case, the length of the midsegment is given as 6x – 24.
The average of the lengths of the bases is calculated by adding the lengths of the bases and dividing the sum by 2. In this case, we have a base length of 5 and another base length of 2x + 7. So, the average of the lengths of the bases is:
(5 + 2x + 7) / 2
We can set up an equation by equating the average of the lengths of the bases to the length of the midsegment. It looks like this:
(5 + 2x + 7) / 2 = 6x – 24
To solve this equation, we need to simplify and isolate the variable x. Let's start by simplifying the equation:
(2x + 12) / 2 = 6x – 24
Next, let's remove the fraction by multiplying both sides of the equation by 2:
2x + 12 = 12x – 48
Now, let's simplify and solve for x. We'll start by moving the variables to one side and constants to the other side:
12 – 2x = 12x – 48
Combine like terms:
14x = 60
Finally, isolate x by dividing both sides of the equation by 14:
x = 60 / 14
Now, let's simplify the fraction:
x ≈ 4.2857
So, the value of x is approximately 4.2857.