Question

the center of a circle is at the origin on a coordinate grid. a line with a positive slope intersects the circle at (0, 7). statement must be true?
the circle has a radius greater than 7.
the circle has a radius equal to 7.
the slope of the line is equal to 7.
the slope of the line is not equal to 7.

Explanation:

Step1: Calculate the radius of the circle

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). The center of the circle is \((0,0)\) and a point on the circle is \((0,7)\). Substituting into the formula: \(r=\sqrt{(0 - 0)^2+(7 - 0)^2}=\sqrt{0 + 49}=7\). So the radius of the circle is \(7\), which means the first option (The circle has a radius greater than \(7\)) is wrong and the second option (The circle has a radius equal to \(7\)) is correct in terms of radius calculation.

Step2: Analyze the slope of the line

The slope formula of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Since we only know one point \((0,7)\) on the line (we don't know the second - point to calculate the exact slope value). But if the slope \(m = 7\), and using the point - slope form \(y - y_1=m(x - x_1)\) (with \((x_1,y_1)=(0,7)\)), the equation of the line is \(y-7 = 7(x - 0)\) or \(y=7x + 7\). The slope of a non - vertical line is \(m=\frac{\Delta y}{\Delta x}\). A slope of \(7\) means for a change in \(x\) by \(1\), \(y\) changes by \(7\). But we can also have other lines with positive slopes passing through \((0,7)\) (e.g., \(y=x + 7\) has a slope of \(1\)). The general form of the slope of a line passing through \((0,7)\) and \((x,y)\) is \(m=\frac{y - 7}{x-0}\) (\(x
eq0\)). Since we don't have enough information to say the slope is \(7\), but we can say that the slope of a non - vertical line is \(m=\frac{\Delta y}{\Delta x}\) and a slope of \(7\) is just one of the many possible values for the slope of a line passing through \((0,7)\) (a line with slope \(7\) is \(y=7x + 7\), but there are lines like \(y = 2x+7\) etc.). So the statement "The slope of the line is not equal to \(7\)" is not necessarily true (because the line could have a slope of \(7\)), and the statement "The slope of the line is equal to \(7\)" is also not necessarily true.

Answer:

The circle has a radius equal to \(7\).