Area on the Coordinate Plane — Practice Set 2

Area on the Coordinate Plane

Practice Set 2 — Absolute Values, Circles & Inscribed Shapes

Find the area enclosed by \[ y = -|x-3| + 8 \quad\text{and}\quad y = |x-3| - 4. \]
What is the area of the region bounded by \[ y = 10 - |x+2| ? \]
Find the area of the region described by \[ |x-5| + |y+4| = 12. \]
What is the area enclosed by \[ 4|x| + |y| = 16? \]
Find the absolute difference between the lengths of the diagonals of \[ 5|x+1| + 2|y-2| = 20. \]
Find the area enclosed by the circle \[ (x+6)^2 + (y-2)^2 = 49. \] Express your answer in terms of \(\pi\).
The area enclosed by \[ |x-4| + |y+1| = b \] is \(98\) square units. Find \(b\).
If the region defined by \[ 3|x-2| + 6|y| = k \] has an area of \(72\), find the value of \(k\).
A circle is inscribed in the diamond \[ 9|x| + 12|y| = 108. \] Find the area of the circle in terms of \(\pi\).
A square is inscribed in the circle \[ (x-1)^2 + (y+5)^2 = 50. \] What is the area of the square?
Find the area enclosed by \[ y = |x-7| - 2 \quad\text{and}\quad y = -|x-7| + 10. \]
What is the area enclosed by \[ |x+3| + 2|y| = 18? \]
A diamond described by \[ 3|x-2| + 4|y-1| = p \] has diagonals whose lengths differ by \(10\). Find \(p\).
The circle \[ (x-8)^2 + (y+3)^2 = 25 \] has a regular hexagon inscribed in it. Find the area of the hexagon in terms of \(\pi\).
A rectangle with sides parallel to the coordinate axes is inscribed in \[ |x| + |y| = 20. \] What is the maximum possible area of the rectangle? Justify your answer.

\(|x-h| + |y-k| = R\) represents 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: \(\pi r^2\).
Area of a square inscribed in a circle of radius \(r\): \(2r^2\).
Area of a regular hexagon inscribed in a circle of radius \(r\): \(\dfrac{3\sqrt{3}}{2}r^2\).

Use diagrams where helpful and clearly label all key points.