Question

a line that contains the points a(-3,8) and b(4,1). use the grid below, along with a ruler and compass, to help locate the point or points.
exercise #5: would the portion of $overline{ab}$ contained inside of the circle be longer or shorter than the diameter of the circle? explain.

Explanation:

Step1: Find the equation of the line passing through A and B

The slope $m$ of the line passing through $A(-3,8)$ and $B(4,1)$ is $m=\frac{y_B - y_A}{x_B - x_A}=\frac{1 - 8}{4+3}=\frac{-7}{7}=-1$.
Using the point - slope form $y - y_1=m(x - x_1)$ with point $A(-3,8)$, we have $y - 8=-1(x + 3)$, which simplifies to $y=-x + 5$.

Step2: Find the equation of the circle

The equation of a circle with center $(h,k)=(2,4)$ and radius $r = 5$ is $(x - 2)^2+(y - 4)^2=25$.

Step3: Substitute $y=-x + 5$ into the circle's equation

$(x - 2)^2+((-x + 5)-4)^2=25$.
$(x - 2)^2+(-x + 1)^2=25$.
Expand: $x^{2}-4x + 4+x^{2}-2x + 1=25$.
Combine like - terms: $2x^{2}-6x-20 = 0$.
Divide by 2: $x^{2}-3x - 10=0$.
Factor: $(x - 5)(x+2)=0$.
Solve for $x$: $x = 5$ or $x=-2$.

Step4: Find the corresponding $y$ values

When $x = 5$, $y=-5 + 5=0$.
When $x=-2$, $y=-(-2)+5=7$.
So the intersection points are $(5,0)$ and $(-2,7)$.

For the second part:

Step1: Find the distance between A and B

The distance $d=\sqrt{(x_B - x_A)^2+(y_B - y_A)^2}=\sqrt{(4 + 3)^2+(1 - 8)^2}=\sqrt{49 + 49}=\sqrt{98}\approx9.9$.
The diameter of the circle $D = 2r=10$.

Step2: Analyze the length of the portion of $\overline{AB}$ inside the circle

We can use geometric reasoning or calculate the intersection points of the line and the circle and then find the length of the segment of the line inside the circle. Another way is to note that the distance from the center of the circle $(2,4)$ to the line $y=-x + 5$ (or $x+y - 5=0$) is $d_{center - line}=\frac{|2 + 4-5|}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt{2}}\approx0.71$.
We know that the circle and the line intersect at two points. By visualizing or calculating the lengths of the sub - segments of $\overline{AB}$ inside the circle, we can see that the length of the portion of $\overline{AB}$ inside the circle is shorter than the diameter of the circle.

Answer:

The intersection points of the circle and the line are $(5,0)$ and $(-2,7)$. The portion of $\overline{AB}$ contained inside of the circle is shorter than the diameter of the circle.