Question

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

options:
4 = (1/2)(x + 7).
x = (1/2)(4 + 7).
7 = (1/2)(8 + 4) + (7 + 14).
7 = (1/2)(x + 4).

Explanation:

Step1: Recall Midline Theorem

In a trapezoid, the midline (or midsegment) is parallel to the two bases and its length is the average of the lengths of the two bases. The formula for the midline \( m \) of a trapezoid with bases \( a \) and \( b \) is \( m=\frac{1}{2}(a + b) \).

Here, the segment of length 7 is the midline, and the two bases are of lengths 4 and \( x \). So according to the midline theorem, \( 7=\frac{1}{2}(x + 4) \).

Step2: Verify with \( x = 10 \)

Substitute \( x = 10 \) into the equation \( 7=\frac{1}{2}(x + 4) \):
Right - hand side: \( \frac{1}{2}(10 + 4)=\frac{1}{2}\times14 = 7 \), which is equal to the left - hand side. So this equation verifies that \( x = 10 \).

Let's check other options:

• For \( 4=\frac{1}{2}(x + 7) \), substitute \( x = 10 \): \( \frac{1}{2}(10+7)=\frac{17}{2}=8.5

eq4 \).

• For \( x=\frac{1}{2}(4 + 7) \), \( \frac{1}{2}(11)=5.5

eq10 \).

• For \( 7=\frac{1}{2}(8 + 4)+(7 + 14) \), \( \frac{1}{2}\times12+21=6 + 21 = 27

eq7 \).

Answer:

\( 7=\frac{1}{2}(x + 4) \) (the last option in the list)