Question

1. which equation verifies that the value of \\(x\\) in the diagram is equal to 10?

• \\(7 = \frac{1}{2}(x + 4)\\)
• \\(7 = \frac{1}{2}(x - 4)\\)
• \\(x = \frac{1}{2}(4 + 7)\\)
• \\(4 = \frac{1}{2}(x + 7)\\)

Explanation:

🆕 New Concept Discovered: Trapezoid Midsegment Theorem
The segment connecting the midpoints of the legs is parallel to the bases and its length is the average of the base lengths.

Step 1: Identify the geometric properties

The diagram shows a trapezoid with:

• A top base of length \( 4 \)
• A bottom base of length \( x \)
• A segment of length \( 7 \) connecting the non-parallel sides

The tick marks on the left leg and the double tick marks on the right leg indicate that this segment connects the midpoints of both legs. Therefore, this segment is the midsegment of the trapezoid.

Step 2: Apply the Midsegment Theorem

The Trapezoid Midsegment Theorem states that the length of the midsegment is equal to half the sum of the lengths of the bases:

$$ \text{Midsegment} = \frac{1}{2}(\text{Base}_1 + \text{Base}_2) $$

Substituting the given values:

$$ 7 = \frac{1}{2}(4 + x) $$

Step 3: Verify the value of \( x \)

We can solve the equation to verify if \( x = 10 \):

$$ 7 = \frac{1}{2}(4 + 10) $$

$$ 7 = \frac{1}{2}(14) $$

$$ 7 = 7 $$

This confirms that the equation \( 7 = \frac{1}{2}(4 + x) \) correctly verifies that \( x = 10 \).

Answer:

The correct equation is:

$$ 7 = \frac{1}{2}(4 + x) $$

(This corresponds to the second option: \( 7 = \frac{1}{2}(4 + x) \))