If points (0,4), (0, -3), (7, -3), and (j, 4) are consecutive vertices of a trapezoid of area 35, what is the value of j ?

Options

A. 3
B. 4
C. 7
D. 10

Explanation

To find the value of \( j \) for the trapezoid formed by the points \( (0, 4) \), \( (0, -3) \), \( (7, -3) \), and \( (j, 4) \) with an area of 35, we can follow these steps:

### Step 1: Identify the parallel sides
The points \( (0, 4) \) and \( (j, 4) \) are on the line \( y = 4 \), and the points \( (0, -3) \) and \( (7, -3) \) are on the line \( y = -3 \). Thus, the lengths of the parallel sides are:
- Length of the top side (between \( (0, 4) \) and \( (j, 4) \)): \( |j - 0| = j \)
- Length of the bottom side (between \( (0, -3) \) and \( (7, -3) \)): \( |7 - 0| = 7 \)

### Step 2: Calculate the height of the trapezoid
The height of the trapezoid is the vertical distance between the two parallel sides:
- Height = \( 4 - (-3) = 4 + 3 = 7 \)

### Step 3: Use the area formula for trapezoids
The area \( A \) of a trapezoid can be calculated using the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \( b_1 \) and \( b_2 \) are the lengths of the parallel sides, and \( h \) is the height.

Substituting the known values:
\[ 35 = \frac{1}{2} \times (j + 7) \times 7 \]

### Step 4: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 70 = (j + 7) \times 7 \]
Now divide both sides by 7:
\[ 10 = j + 7 \]

### Step 5: Solve for \( j \)
To find \( j \), subtract 7 from both sides:
\[ j = 10 - 7 \]
\[ j = 3 \]

### Final Answer
Thus, the value of \( j \) is \( 3 \).

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