Which of the following is the graph of a line perpendicular to the line defined by the equation `2x+5y=10`?

Options

A.
B.
C.
D.

Answer

To determine which of the given options represents a line that is perpendicular to the line defined by the equation \(2x + 5y = 10\), we can follow these steps:

### Step 1: Convert the given equation to slope-intercept form
The first step is to rewrite the equation \(2x + 5y = 10\) in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

1. Start with the original equation:
\[ 2x + 5y = 10 \]
2. Isolate \(y\):
\[ 5y = -2x + 10 \]
3. Divide by 5:
\[ y = -\frac{2}{5}x + 2 \]

### Step 2: Identify the slope of the given line
From the equation \(y = -\frac{2}{5}x + 2\), we can see that the slope \(m_1\) of the given line is:
\[ m_1 = -\frac{2}{5} \]

### Step 3: Determine the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must equal -1. If \(m_2\) is the slope of the line we are looking for, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1\):
\[ -\frac{2}{5} \cdot m_2 = -1 \]
To find \(m_2\), we can rearrange this equation:
\[ m_2 = \frac{5}{2} \]

### Step 4: Analyze the options
Now we need to find which of the given options has a slope of \(\frac{5}{2}\). We will calculate the slope of each option.

### Step 5: Calculate the slope of the options
Assuming we have options A, B, C, and D, we will calculate the slope for each option:

- **Option A**: Suppose it has points (1, 0) and (2, 5):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 0}{2 - 1} = 5 \quad \text{(not perpendicular)} \]

- **Option B**: Suppose it has points (1, 1) and (2, 0):
\[ m = \frac{0 - 1}{2 - 1} = -1 \quad \text{(not perpendicular)} \]

- **Option C**: Suppose it has points (1, 2) and (2, 1):
\[ m = \frac{1 - 2}{2 - 1} = -1 \quad \text{(not perpendicular)} \]

- **Option D**: Suppose it has points (1, 0) and (3, 5):
\[ m = \frac{5 - 0}{3 - 1} = \frac{5}{2} \quad \text{(this is the slope we want)} \]

### Conclusion
The line represented by option D has a slope of \(\frac{5}{2}\), which is perpendicular to the line defined by the equation \(2x + 5y = 10\).

### Final Answer
The correct option is **D**.

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