Mathematics Admissions Test (MAT) Geometry and Coordinate Geometry Syllabus

Geometry and Coordinate Geometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Geometry and Coordinate Geometry in Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Coordinate Geometry of Lines

   3 topics

   • Straight Lines
     • Gradient, intercept and equation forms
     • Parallel and perpendicular lines
     • Distance between points and midpoint
   • Intersections and Regions
     • Point of intersection of two lines
     • Regions defined by linear inequalities
   • Distance from a Point to a Line
     • Perpendicular distance and foot of perpendicular
2. 2

   Circles

   3 topics

   • Equation of a Circle
     • Centre-radius form and the general form
     • Completing the square to find centre and radius
   • Lines and Circles
     • Tangents, chords and the perpendicular from the centre
     • Intersection of a line and a circle
   • Circle Theorems
     • Angle in a semicircle and tangent-radius perpendicularity
3. 3

   Plane Geometry and Mensuration

   3 topics

   • Areas and Lengths
     • Arc length and sector area in radians
     • Areas of composite plane figures
   • Similarity and Pythagoras
     • Similar triangles and ratio reasoning
     • Pythagoras in 2D and 3D
   • Geometric Problem Solving
     • Combining coordinate and pure geometry methods

Geometry and Coordinate Geometry flashcards for Mathematics Admissions Test (MAT)

23 of 50 cards from the Geometry and Coordinate Geometry deck — real questions with worked answers.

1. What is the gradient (slope) of the straight line passing through points $(x_1, y_1)$ and $(x_2, y_2)$?

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ provided $x_1 \neq x_2$. It measures the change in $y$ per unit change in $x$.

2. State the gradient-intercept form of the equation of a straight line and name its parameters.

   $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept (the value of $y$ where the line crosses the $y$-axis at $x = 0$).

3. Give the point-slope form of a straight line through the point $(x_1, y_1)$ with gradient $m$.

   $$y - y_1 = m(x - x_1)$$ This is the most efficient form when you know one point and the gradient.

4. What is the relationship between the gradients of two parallel lines, and of two perpendicular lines?

   Parallel lines have equal gradients: $m_1 = m_2$. Perpendicular lines have gradients whose product is $-1$: $m_1 m_2 = -1$, i.e. $m_2 = -\frac{1}{m_1}$.

5. What is the general (standard) form of a line's equation, and how do you read off its gradient?

   $ax + by + c = 0$. The gradient is $m = -\frac{a}{b}$ (for $b \neq 0$), and the $y$-intercept is $-\frac{c}{b}$.

6. Describe the gradients of a horizontal line and a vertical line, and give their equation forms.

   A horizontal line has gradient $0$ and equation $y = k$. A vertical line has an undefined (infinite) gradient and equation $x = k$, where $k$ is a constant.

7. How do you find the coordinates of the point where two non-parallel lines intersect?

   Solve their two equations simultaneously (by substitution or elimination). The simultaneous solution $(x, y)$ is the unique intersection point. Parallel lines give no solution; identical lines give infinitely many.

8. How does the inequality $y > mx + c$ (strict) divide the plane into regions, and how is it shown graphically?

   The line $y = mx + c$ is the boundary; $y > mx + c$ is the region above it. A strict inequality ($<$ or $>$) uses a dashed line (boundary excluded); $\leq$ or $\geq$ uses a solid line (boundary included).

9. When sketching the region satisfying several simultaneous inequalities, what region do you shade/keep?

   The feasible region is the intersection (overlap) of all the individual half-planes — the set of points satisfying every inequality at once. It is bounded by the relevant line segments.

10. How can you quickly test which side of a line $ax + by + c = 0$ a region lies on?

    Substitute a convenient test point not on the line (often the origin $(0,0)$) into the inequality. If it satisfies the inequality, that point's side is the required region; otherwise it is the other side.

11. State the formula for the perpendicular distance from a point $(x_0, y_0)$ to the line $ax + by + c = 0$.

    $$d = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}$$ The absolute value ensures a non-negative distance.

12. Using the distance-to-line formula, what is the distance from the origin to the line $ax + by + c = 0$?

    $$d = \frac{|c|}{\sqrt{a^2 + b^2}}$$ obtained by substituting $(x_0, y_0) = (0,0)$.

13. What is the geometric meaning of the perpendicular distance from a point to a line, and how would you derive it without the formula?

    It is the shortest distance, measured along the line through the point perpendicular to the given line. Derive it by finding the foot of the perpendicular (intersection of the line with the perpendicular through the point) and applying the distance formula between the two points.

14. State the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane.

    $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This follows directly from Pythagoras' theorem.

15. Give the coordinates of the midpoint of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$.

    $$\left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$$ the average of the coordinates.

16. State the standard (centre-radius) equation of a circle with centre $(a, b)$ and radius $r$.

    $$(x - a)^2 + (y - b)^2 = r^2$$

17. What is the equation of a circle of radius $r$ centred at the origin?

    $$x^2 + y^2 = r^2$$

18. Given the general circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$, state its centre and radius.

    Centre $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$ (real circle requires $g^2 + f^2 - c > 0$).

19. What technique converts $x^2 + y^2 + Dx + Ey + F = 0$ into centre-radius form?

    Completing the square in $x$ and in $y$: $\left(x + \frac{D}{2}\right)^2 + \left(y + \frac{E}{2}\right)^2 = \frac{D^2}{4} + \frac{E^2}{4} - F$, giving centre $\left(-\frac{D}{2}, -\frac{E}{2}\right)$.

20. How do you find where a line meets a circle algebraically, and what do the number of solutions tell you?

    Substitute the line's equation into the circle's equation to get a quadratic. Two real roots = line is a secant (two intersection points); one repeated root = tangent (one point); no real roots = line misses the circle. Use the discriminant to decide.

21. What is the key geometric property of a tangent to a circle at the point of contact?

    The tangent is perpendicular to the radius drawn to the point of contact. Hence the radius and tangent meet at $90^\circ$.

22. To prove a line is tangent to a circle, what condition on the distance from the centre must hold?

    The perpendicular distance from the circle's centre to the line must equal the radius: $d = r$. If $d < r$ it is a secant; if $d > r$ the line misses the circle.

23. How do you find the equation of the tangent to a circle at a known point $P$ on it?

    Find the gradient of the radius from the centre to $P$, take the negative reciprocal for the tangent's gradient, then use point-slope form through $P$. The tangent is perpendicular to that radius.

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