1. The problem is to identify the correct formula for the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$. Recall the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Looking at the options, option D matches this formula. 2. Find the midpoint between points $S(2,7)$ and $T(8,-1)$. Midpoint formula: $$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ Calculate each coordinate: $$x = \frac{2 + 8}{2} = 5,$$ $$y = \frac{7 + (-1)}{2} = \frac{6}{2} = 3.$$ Midpoint is $(5,3)$. 3. Find the midpoint between points $M(-4, -10)$ and $N(4,7)$. Using the midpoint formula: $$x = \frac{-4 + 4}{2} = 0,$$ $$y = \frac{-10 + 7}{2} = \frac{-3}{2} = -\frac{3}{2}.$$ Midpoint is $(0,-\frac{3}{2})$. 4. Find the distance between points $A(0,0)$ and $B(4,3)$. Distance formula: $$d = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.$$ 5. Find the distance between points $C(2,1)$ and $D(5,-3)$. Calculate: $$d = \sqrt{(5 - 2)^2 + (-3 - 1)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$$ 6. Each unit equals 2 km. Distance from policeman at $(4,0)$ to car at $(1,-4)$: First find distance in units: $$d = \sqrt{(4-1)^2 + (0 - (-4))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5.$$ Convert to km: $$5 \times 2 = 10\text{ km}.$$ 7. Find the center of the circle equation $$(x + 2)^2 + (y - 7)^2 = 10.$$ Center is at $$(-2, 7).$$ 8. Find the radius of the circle $$x^2 + y^2 = 169.$$ Radius is $$\sqrt{169} = 13.$$ 9. Write the equation for a circle center $(-3,6)$ with radius $12$. Standard form: $$ (x - h)^2 + (y - k)^2 = r^2 $$ Substitute: $$ (x + 3)^2 + (y - 6)^2 = 144.$$ Expanding: $$ x^2 + 6x + 9 + y^2 - 12y + 36 = 144 $$ Simplify: $$ x^2 + y^2 + 6x - 12y + 45 = 144 $$ $$ x^2 + y^2 + 6x - 12y = 99.$$ Matches option B. 10. Center at (1, -1) and radius is positive length, so $r=2$. Answer is D. 11. The tower is at $(5,7)$ with radius 10 km. Standard equation: $$(x - 5)^2 + (y - 7)^2 = 10^2 = 100.$$ Answer is C. 12. Distance from Stell's house $(2,3)$ to tower $(5,7)$: $$ d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = 5.$$ 13. Distance from Pablo's house $(7,4)$ to tower $(5,7)$: $$ d = \sqrt{(7 - 5)^2 + (4 - 7)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6.$$ Since the radius is 10 km, both houses are within range. Answer is B. 14. Distance between cars A$(2,3)$ and B$(7,15)$: $$ d = \sqrt{(7 - 2)^2 + (15 - 3)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13.$$ 15. Midpoint between the two cars: $$ \left( \frac{2 + 7}{2}, \frac{3 + 15}{2} \right) = (\frac{9}{2}, 9) = (4.5, 9).$$ Option B matches $(6,9)$ which is incorrect. But option A shows $(37/2, 9)$ which is $18.5$, also incorrect. None of the options exactly matches the midpoint $(4.5,9)$, closest is B but it is not correct. But from the given options, B (6,9) is the best approximate choice assuming a typo. Final answers: 1: D 2: C 3: D 4: C 5: C 6: C 7: B 8: A 9: B 10: D 11: C 12: B 13: B 14: C 15: B