7.3 Circles & Coordinate Geometry

In the Quantitative Reasoning section of the Saudi Arabia Qiyas General Aptitude Test (GAT), circles and coordinate geometry are highly structured topics, frequently combined to test analytical reasoning and spatial relationships. Shaded region problems involving circles are a staple of the exam, demanding speed and a clear understanding of area subtraction. Meanwhile, coordinate geometry questions focus on distance, midpoints, slopes, and linear relationships on the Cartesian plane.

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Circle Geometry: Basics and Formulas

A circle is defined as all points in a plane that are equidistant from a center point.

Fundamental Formulas:

1. Circumference ($C$): The perimeter of a circle. $C = 2\pi r = \pi d$ where $r$ is the radius and $d$ is the diameter ($d = 2r$).
2. Area ($A$): $A = \pi r^2$
3. Arc Length ($L$): An arc is a portion of the circumference. The length of an arc subtended by a central angle $\theta$ (in degrees) is: $L = \frac{\theta}{360^\circ} \times 2\pi r$
4. Sector Area ($A_{\text{sector}}$): A sector is a slice of the circle (like a pizza slice). The area of a sector with central angle $\theta$ is: $A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$

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Essential Circle Theorems for GAT

Qiyas GAT tests several geometric theorems regarding angles and lines in circles:

1. Central vs. Inscribed Angles:

• A central angle has its vertex at the center of the circle. Its measure is equal to the measure of its intercepted arc.
• An inscribed angle has its vertex on the circle. Its measure is exactly half of the measure of its intercepted arc, and thus half of the central angle that intercepts the same arc.
• Corollary: Inscribed angles that intercept the same arc are equal.

1. Angle Inscribed in a Semicircle: Any inscribed angle that intercepts a diameter (or semicircle) is exactly $90^\circ$. If you see a triangle drawn inside a circle where one side is the diameter and the opposite vertex lies on the circle, it is automatically a right-angled triangle, and the diameter is the hypotenuse. This is a common setup for combining circle formulas with the Pythagorean theorem.

2. Tangents to a Circle:

• A tangent line touches the circle at exactly one point.
• Tangent-Radius Theorem: A tangent is perpendicular to the radius drawn to the point of tangency (forming a $90^\circ$ angle).
• Tangent Segments: Tangent segments drawn to a circle from the same external point are equal in length.

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Shaded Region and Overlapping Figure Problems

The GAT is famous for composite geometry problems, particularly shaded regions involving circles and polygons. Solving these requires breaking the diagram down into recognizable parts and performing subtractions.

Common Configurations:

• Circle Inscribed in a Square:
  • If a circle is inscribed in a square of side length $s$, the diameter of the circle is equal to the side of the square ($d = s$, so $r = \frac{s}{2}$).
  • Area of Square = $s^2$
  • Area of Circle = $\pi r^2 = \pi \left(\frac{s}{2} ight)^2 = \frac{\pi s^2}{4}$
  • Shaded Area (Corners): $A_{\text{shaded}} = s^2 - \frac{\pi s^2}{4} = s^2\left(1 - \frac{\pi}{4} ight)$.
• Square Inscribed in a Circle:
  • If a square is inscribed in a circle of radius $r$, the diagonal of the square is equal to the diameter of the circle ($d_{\text{square}} = 2r$).
  • Using the diagonal formula: $\text{Area of Square} = \frac{d^2}{2} = \frac{(2r)^2}{2} = 2r^2$.
  • Area of Circle = $\pi r^2$
  • Shaded Area (Segments): $A_{\text{shaded}} = \pi r^2 - 2r^2 = r^2(\pi - 2)$.

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Coordinate Geometry on the Cartesian Plane

Coordinate geometry translates geometric figures into algebraic equations on a coordinate plane with an $x$-axis and a $y$-axis divided into four quadrants.

Core Formulas:

1. Distance Formula: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
2. Midpoint Formula: The midpoint coordinates $M$ between $(x_1, y_1)$ and $(x_2, y_2)$ are: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ GAT Application: Finding the center of a circle when given the coordinates of the endpoints of its diameter.
3. Slope ($m$): The steepness of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Have equal slopes ($m_1 = m_2$).
• Perpendicular Lines: Have negative reciprocal slopes ($m_1 \cdot m_2 = -1$ or $m_2 = -\frac{1}{m_1}$).

1. Equation of a Line (Slope-Intercept Form): $y = mx + b$ where $m$ is the slope and $b$ is the $y$-intercept.
2. Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center of the circle and $r$ is the radius.

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Qiyas GAT Common Traps and Exam Strategies

• The Radius vs. Diameter Trap: Always check if the question gives the radius or the diameter. A common mistake is to plug the diameter directly into $\pi r^2$.
• Approximating $\pi$: In Qiyas GAT, options are usually left in terms of $\pi$ (e.g., $16\pi$). If the options contain decimals, use $\pi \approx 3.14$ or $\pi \approx \frac{22}{7}$ (especially if the radius is a multiple of 7).
• Coordinate Quadrant Signs: Pay close attention to negative signs when calculating slopes or distances. The signs change depending on the quadrant: Quadrant I $(+, +)$, Quadrant II $(-, +)$, Quadrant III $(-, -)$, Quadrant IV $(+, -)$.
• Grid Drawing Shortcut: If a coordinate geometry question is abstract, quickly sketch a rough Cartesian grid on your scratch paper to visualize the points. This often reveals the answer without executing full formulas.