In the figure, 𝐿𝐻 is the midsegment of trapezoid 𝐹𝐺𝐽𝐾. What is the value of 𝑥?
So we have a diagram of a trapezoid. And we’re told that the line segment 𝐿𝐻 is its midsegment. What does this is mean? Firstly, the midsegment of a trapezoid is parallel to each base. So in this case, that’s the lines 𝐹𝐺 and 𝐾𝐽.
Secondly, the midsegment connects the midpoints of the legs of the trapezoid. We can see the 𝐿 and 𝐻 are the midpoints of the two legs because the line segments 𝐹𝐿 and 𝐿𝐾 are equal in length and also the line segments 𝐺𝐻 and 𝐻𝐽 are equal to each other in length.
We’ve been told the length of one of the bases of the trapezoid 𝐾𝐽 and the length of the midsegment. The value we’re looking to calculate, 𝑥, is the length of the second base of the trapezoid 𝐹𝐺.
In order to answer this question, we need to recall the trapezoid midsegment theorem. It tells us this. The measure of the midsegment of a trapezoid is half the sum of the lengths of the bases. In this trapezoid, the bases are 23.6 and 𝑥. The length of the midsegment is 18.6. So we can express this as an equation. Half the sum of the length of the bases, so that’s 𝑥 plus 23.6 over two, is equal to 18.6.
To find 𝑥, we now need to solve this equation. The first step is to multiply both sides of the equation by two. This gives 𝑥 plus 23.6 is equal to 37.2. The final step in solving this equation is to subtract 23.6 from both sides. This gives 𝑥 is equal to 13.6. So we found the value of 𝑥. It’s 13.6.
Remember, the key fact we used in this question was the trapezoid midsegment theorem, which tells us that the measure of the midsegment of a trapezoid is half the sum of the lengths of the bases.