ACT Math || SAT Math || Solution of Area on the Coordinate Plane (Questions published on Dec 8, 2025)

Area on the Coordinate Plane

Full Step-by-Step Solutions (Problems 1–15)

Working Conventions / Reminders

• For \( |x-h|+|y-k|=R \), the graph is a diamond with area \(2R^2\).
• For \( a|x-h| + b|y-k| = c \), diagonals are \(2c/a\) and \(2c/b\).
• Area of a circle: \(A=\pi r^2\).

Solutions

1. Problem 1
   $$ y = -|x+4| + 7 $$
   $$ y = |x+4| - 5 $$
   Set equal:

   $$ -|x+4|+7 = |x+4|-5 $$

   $$ 12 = 2|x+4| $$

   $$ |x+4|=6 $$

   $$ x=-10,\ 2 $$
   $$ A=\frac{12\times12}{2}=72 $$

   $$ \boxed{72} $$
2. Problem 2
   $$ y = 12-|x-6| $$
   Set \(y=0\):

   $$ 12-|x-6|=0 $$

   $$ |x-6|=12 $$

   $$ x=-6,\ 18 $$
   $$ A=\frac12(24)(12)=144 $$

   $$ \boxed{144} $$
3. Problem 3
   $$ |x-8|+|y-1|=10 $$
   $$ A=2(10)^2=200 $$

   $$ \boxed{200} $$
4. Problem 4
   $$ 3|x|+2|y|=18 $$
   $$ A=\frac{12\times18}{2}=108 $$

   $$ \boxed{108} $$
5. Problem 5
   $$ 4|x-2|+|y+3|=20 $$
   $$ |40-10|=30 $$

   $$ \boxed{30} $$
6. Problem 6
   $$ (x-4)^2+(y-9)^2=25 $$
   $$ A=25\pi $$

   $$ \boxed{25\pi} $$
7. Problem 7
   $$ 2b^2=72 $$
   $$ b=6 $$

   $$ \boxed{6} $$
8. Problem 8
   $$ 6|x+1|+3|y-5|=k $$
   $$ k=9\sqrt6 $$

   $$ \boxed{9\sqrt6} $$
9. Problem 9
   $$ 7|x|+24|y|=84 $$
   $$ A=\frac{7056\pi}{625} $$

   $$ \boxed{\frac{7056\pi}{625}} $$
10. Problem 10
    $$ (x-3)^2+(y+4)^2=32 $$
    $$ A=64 $$

    $$ \boxed{64} $$
11. Problem 11
    $$ y = |x+10|-3 $$
    $$ y = -|x+10|+9 $$
    $$ A=\frac{12\times12}{2}=72 $$

    $$ \boxed{72} $$
12. Problem 12
    $$ |x-2|+3|y|=15 $$
    $$ A=\frac{30\times10}{2}=150 $$

    $$ \boxed{150} $$
13. Problem 13
    $$ 2|x-1|+5|y+6|=p $$
    $$ p=10 $$

    $$ \boxed{10} $$
14. Problem 14
    $$ (x+12)^2+(y-1)^2=18 $$
    $$ A=27\sqrt3 $$

    $$ \boxed{27\sqrt3} $$
15. Problem 15
    $$ |x|+|y|=14 $$
    $$ x=y=7 $$

    $$ A=196 $$

    $$ \boxed{196} $$

Quick Answers

1. 72
2. 144
3. 200
4. 108
5. 30
6. \(25\pi\)
7. 6
8. \(9\sqrt6\)
9. \(\frac{7056\pi}{625}\)
10. 64
11. 72
12. 150
13. 10
14. \(27\sqrt3\)
15. 196