Question

1. in order for a circle to intersect s, t, and u, it must contain s and t. 2. ? 3. let (g, 6) be the center of the circle containing s and t. 4. the distance between (g, 6) and u is √9+(5 - g)². 5. the distance between (g, 6) and t is √4+(3 - g)². 6. the solution to √9+(5 - g)²=√4+(3 - g)² is g = 5.25.

Explanation:

Step1: Recall distance - formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Analyze the given equation in question 6

We are given the equation \(\sqrt{9+(5 - g)^2}=\sqrt{4+(3 - g)^2}\).
Square both sides to get rid of the square - roots:
\[9+(5 - g)^2=4+(3 - g)^2\]

Step3: Expand the squares

Expand \((5 - g)^2=25-10g + g^2\) and \((3 - g)^2 = 9-6g+g^2\).
The equation becomes \(9 + 25-10g+g^2=4 + 9-6g+g^2\).

Step4: Simplify the equation

Combine like - terms:
\[34-10g+g^2=13-6g+g^2\]
Subtract \(g^2\) from both sides:
\[34-10g=13-6g\]
Add \(10g\) to both sides: \(34 = 13-6g + 10g\).
\[34=13 + 4g\]
Subtract 13 from both sides: \(4g=34 - 13=21\).

Step5: Solve for \(g\)

Divide both sides by 4: \(g=\frac{21}{4}=5.25\).

Answer:

\(g = 5.25\)