1. **Problem:** Which expression represents the distance $d$ between the two points $(x_1,y_1)$ and $(x_2,y_2)$? 1. **Solution:** The horizontal change is $\Delta x = x_2 - x_1$. The vertical change is $\Delta y = y_2 - y_1$. By the Pythagorean theorem the distance is $d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$. Substituting gives $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Therefore the correct choice is C. 2. **Problem:** Point $L$ is the midpoint of $\overline{KM}$. Which is true about the distances among $K$, $L$, and $M$? 2. **Solution:** By definition a midpoint divides a segment into two equal parts. Thus $KL = LM$. Therefore the correct choice is C. 3. **Problem:** San Vicente is at $(4,9)$ and is 13 units from San Luis. Which coordinate could be San Luis? 3. **Solution:** Compute distances to each option using $d=\sqrt{(x-4)^2+(y-9)^2}$. Option A: $(-13,0)$ gives $d=\sqrt{(-13-4)^2+(0-9)^2}=\sqrt{(-17)^2+(-9)^2}=\sqrt{289+81}=\sqrt{370}\ne13$. Option B: $(16,4)$ gives $d=\sqrt{(16-4)^2+(4-9)^2}=\sqrt{12^2+(-5)^2}=\sqrt{144+25}=\sqrt{169}=13$. Option C: $(4,16)$ gives $d=\sqrt{0^2+7^2}=7\ne13$. Option D: $(0,13)$ gives $d=\sqrt{(-4)^2+4^2}=\sqrt{16+16}=\sqrt{32}\ne13$. Therefore the correct choice is B. 4. **Problem:** What is the distance between $M(-3,1)$ and $N(7,-3)$? 4. **Solution:** Compute differences $\Delta x=7-(-3)=10$ and $\Delta y=-3-1=-4$. Distance $d=\sqrt{10^2+(-4)^2}=\sqrt{100+16}=\sqrt{116}=2\sqrt{29}$. Therefore the correct choice is B. 5. **Problem:** Which represents the midpoint $M$ of endpoints $(x_1,y_1)$ and $(x_2,y_2)$? 5. **Solution:** The midpoint formula averages coordinates giving $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Therefore the correct choice is B. 6. **Problem:** Midpoint of endpoints $(-1,-3)$ and $(11,7)$. 6. **Solution:** Average the coordinates: $x$-midpoint $=\frac{-1+11}{2}=5$. $y$-midpoint $=\frac{-3+7}{2}=2$. Midpoint is $(5,2)$. Therefore the correct choice is D. 7. **Problem:** Which equation describes a circle with radius 4 units? 7. **Solution:** A circle with radius 4 has equation $(x-h)^2+(y-k)^2=4^2=16$ for some center $(h,k)$. Option B and C are identical as $(x-2)^2+(y-2)^2=4^2$, which gives radius 4. Choosing one matching option, the correct choice is B. 8. **Problem:** Points $P(-2,5)$ and $Q(8,5)$ are shown. Which expression gives the distance between them? 8. **Solution:** They share the same $y$ so distance is horizontal difference $|8-(-2)|=10$. Evaluate options: A gives $|-2-5|=7$, B gives $8-5=3$, C gives $8-2=6$, D gives $|-2-8|=10$. Therefore the correct choice is D. 9. **Problem:** New tower midway between $(-5,-3)$ and $(9,13)$. What are the coordinates? 9. **Solution:** Midpoint is $\left(\frac{-5+9}{2},\frac{-3+13}{2}\right)=\left(\frac{4}{2},\frac{10}{2}\right)=(2,5)$. Therefore the correct choice is A. 10. **Problem:** What proof uses figures on a coordinate plane to prove geometric properties? 10. **Solution:** A coordinate proof places figures in the coordinate plane and uses algebraic calculations. This is called a coordinate proof. Therefore the correct choice is C. 11. **Problem:** Square with vertices $H(3,8),I(15,8),J(15,-4),K(3,-4)$. What is the length of a diagonal? 11. **Solution:** Side length is $15-3=12$. Diagonal of a square is $12\sqrt{2}$. Therefore the correct choice is D. 12. **Problem:** Triangle with $T(-1,-3),O(7,5),P(7,-2)$. Length of segment joining midpoint of $\overline{OT}$ and $P$. 12. **Solution:** Midpoint of $O(7,5)$ and $T(-1,-3)$ is $\left(\frac{7+(-1)}{2},\frac{5+(-3)}{2}\right)=(3,1)$. Distance to $P(7,-2)$ is $\sqrt{(7-3)^2+(-2-1)^2}=\sqrt{4^2+(-3)^2}=\sqrt{16+9}=\sqrt{25}=5$. Therefore the correct choice is A. 13. **Problem:** What figure is formed by $A(3,7),B(11,10),C(11,5),D(3,2)$ connected consecutively? 13. **Solution:** Vectors show $AB=(8,3)$ and $CD=(-8,-3)$ so $AB\parallel CD$ and $BC=(0,-5)$ and $DA=(0,5)$ so $BC\parallel DA$. Opposite sides are parallel, so the quadrilateral is a parallelogram. Therefore the correct choice is A. 14. **Problem:** In parallelogram with $S(0,0),R(b,0),P(a,c)$, find $Q$. 14. **Solution:** In a parallelogram opposite vertices sum: $\vec{Q}=\vec{P}+\vec{R}-\vec{S}$. So $Q=(a+b,c+0)=(a+b,c)$. Therefore the correct choice is B. 15. **Problem:** Distances between Diana $(-3,4)$ and Jolina $(1,7)$. About how far are they? 15. **Solution:** Compute differences $\Delta x=1-(-3)=4$ and $\Delta y=7-4=3$. Distance $=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$. None of the given choices equals 5, so the exact distance is 5 units and the provided options appear incorrect. 16. **Problem:** What is the center of the circle $x^2+y^2-4x+10y+13=0$? 16. **Solution:** Complete the square: $x^2-4x+y^2+10y+13=0$. Add and subtract to form squares: $(x^2-4x+4)+(y^2+10y+25)+13-4-25=0$. Thus $(x-2)^2+(y+5)^2-16=0$ and the center is $(2,-5)$. Therefore the correct choice is C. 17. **Problem:** Point $F$ is 5 units from $D(6,2)$. If $x_F=10$ and $F$ is in the first quadrant, what is $y_F$? 17. **Solution:** Distance equation $\sqrt{(10-6)^2+(y-2)^2}=5$ gives $(y-2)^2+16=25$. So $(y-2)^2=9$ and $y-2=\pm3$ giving $y=5$ or $y=-1$. First quadrant requires $y>0$, so $y=5$. Therefore the correct choice is C. 18. **Problem:** Endpoints of a diameter are $L(-3,-2)$ and $G(9,-6)$. What is the radius length? 18. **Solution:** Diameter length is $\sqrt{(9-(-3))^2+(-6-(-2))^2}=\sqrt{12^2+(-4)^2}=\sqrt{144+16}=\sqrt{160}=4\sqrt{10}$. Radius is half the diameter: $2\sqrt{10}$. Therefore the correct choice is B. 19. **Problem:** A radius has endpoints $(4,-1)$ and $(8,2)$ and the center is in the fourth quadrant. What is the circle equation? 19. **Solution:** The center in the fourth quadrant is $(4,-1)$ since $4>0$ and $-1<0$. Radius squared is $(8-4)^2+(2-(-1))^2=4^2+3^2=16+9=25$. Equation is $(x-4)^2+(y+1)^2=25$. Therefore the correct choice is D. 20. **Problem:** Tower at $(-2,8)$ with transmission radius 12 km. What equation represents the transmission boundary? 20. **Solution:** Center $(-2,8)$ and radius $12$ gives $(x+2)^2+(y-8)^2=144$. Expanding gives $x^2+y^2+4x-16y-76=0$. None of the given options match this expansion exactly, so the correct equation is $(x+2)^2+(y-8)^2=144$ and none of the choices are correct.