To determine how the slopes between the given points compare, let's calculate the slopes step-by-step. Given the points [tex]\((4,30)\)[/tex], [tex]\((10,75)\)[/tex], and [tex]\((12,90)\)[/tex]:
1. Calculating the slope between [tex]\((4,30)\)[/tex] and [tex]\((12,90)\)[/tex]:
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates [tex]\((4, 30)\)[/tex] and [tex]\((12, 90)\)[/tex]:
[tex]\[
m_1 = \frac{90 - 30}{12 - 4} = \frac{60}{8} = 7.5
\][/tex]
2. Calculating the slope between [tex]\((4,30)\)[/tex] and [tex]\((10,75)\)[/tex]:
Using the same slope formula with the coordinates [tex]\((4, 30)\)[/tex] and [tex]\((10, 75)\)[/tex]:
[tex]\[
m_2 = \frac{75 - 30}{10 - 4} = \frac{45}{6} = 7.5
\][/tex]
3. Comparing the two slopes:
From the calculations, we see that:
[tex]\[
m_1 = 7.5 \quad \text{and} \quad m_2 = 7.5
\][/tex]
Therefore, the slopes between the points [tex]\((4,30)\)[/tex] and [tex]\((12,90)\)[/tex], and between [tex]\((4,30)\)[/tex] and [tex]\((10,75)\)[/tex] are indeed the same.
4. Conclusion:
Based on our calculations, the correct statement is:
[tex]\[
\text{The slope between (\(4,30\)) and (\(12,90\)) and between (\(4,30\)) and (\(10,75\)) is the same.}
\][/tex]
Thus, the correct option is:
[tex]\[
\textbf{The slope between (4,30) and (12,90) and between (4,30) and (10,75) is the same.}
\][/tex]
