Mathematics Admissions Test (MAT) Geometry and Coordinate Geometry Syllabus

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 $90^\circ$.

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.

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.

State the 'angle in a semicircle' circle theorem.

The angle subtended at the circumference by a diameter is a right angle ($90^\circ$). Equivalently, the angle in a semicircle is a right angle.

State the 'angle at the centre' circle theorem.

The angle subtended by an arc (or chord) at the centre is twice the angle subtended by the same arc at any point on the major arc (circumference): $\angle\text{centre} = 2 \times \angle\text{circumference}$.

State the 'angles in the same segment' circle theorem.

Angles subtended by the same arc (chord) at the circumference, in the same segment, are equal.

State the cyclic quadrilateral theorem about opposite angles.

Opposite angles of a cyclic quadrilateral (all four vertices on a circle) sum to $180^\circ$: $A + C = 180^\circ$ and $B + D = 180^\circ$.

State the Alternate Segment Theorem.

The angle between a tangent and a chord drawn from the point of contact equals the angle subtended by that chord in the alternate (opposite) segment.

State the two tangents (tangent length) theorem for a circle.

The two tangent segments drawn from an external point to a circle are equal in length, and the line from that external point to the centre bisects the angle between the two tangents.

What does the perpendicular from the centre of a circle to a chord do to that chord?

It bisects the chord (and conversely, the perpendicular bisector of a chord passes through the centre). So a radius perpendicular to a chord meets it at its midpoint.

State the formula for the circumference and the area of a circle of radius $r$.

Circumference $C = 2\pi r$ and area $A = \pi r^2$.

For a sector of angle $\theta$ radians in a circle of radius $r$, give its arc length and area.

Arc length $s = r\theta$ and sector area $A = \frac{1}{2} r^2 \theta$ (with $\theta$ in radians).

For a sector of angle $\theta$ measured in degrees, give the arc length and sector area of a circle radius $r$.

Arc length $= \frac{\theta}{360} \times 2\pi r$ and sector area $= \frac{\theta}{360} \times \pi r^2$.

How do you find the area of the segment of a circle cut off by a chord?

Segment area = sector area $-$ triangle area $= \frac{1}{2} r^2 \theta - \frac{1}{2} r^2 \sin\theta = \frac{1}{2} r^2 (\theta - \sin\theta)$ for angle $\theta$ in radians.

State the area formula for a triangle given two sides $a$, $b$ and the included angle $C$.

$$\text{Area} = \frac{1}{2} a b \sin C$$

State the standard area formula for a triangle, a parallelogram, and a trapezium.

Triangle: $\frac{1}{2} \times \text{base} \times \text{height}$. Parallelogram: $\text{base} \times \text{height}$. Trapezium: $\frac{1}{2}(a + b) h$ where $a, b$ are the parallel sides and $h$ the perpendicular height.

Give the Shoelace formula for the area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$.

$$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

What does it mean for two triangles to be similar, and what follows about their sides and angles?

Similar triangles have the same shape: all corresponding angles are equal and all pairs of corresponding sides are in the same ratio (the scale factor $k$). Same shape, possibly different size.

List the standard conditions sufficient to prove two triangles are similar.

AA (two pairs of equal angles), SSS (all three side ratios equal), and SAS (two side ratios equal with the included angle equal). Any one is sufficient.

If two similar figures have linear scale factor $k$, what are the ratios of their areas and volumes?

Areas scale by $k^2$ and volumes scale by $k^3$. So if lengths double ($k=2$), area $\times 4$ and volume $\times 8$.

State Pythagoras' theorem and its converse.

For a right-angled triangle with hypotenuse $c$: $a^2 + b^2 = c^2$. Converse: if $a^2 + b^2 = c^2$ holds for a triangle's sides, the triangle is right-angled (right angle opposite the longest side $c$).

What is the relationship between similar triangles and the gradient of a straight line?

Any two right-angled 'gradient triangles' drawn under a straight line are similar (AA), so the ratio rise/run is constant along the line — which is precisely why a straight line has a single well-defined gradient.

Define congruent figures and list the congruence conditions for triangles.

Congruent figures are identical in shape and size (scale factor $1$). Triangle congruence conditions: SSS, SAS, ASA, AAS, and RHS (right angle, hypotenuse, side).

State the Sine Rule for a triangle with sides $a, b, c$ opposite angles $A, B, C$.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$ where $R$ is the circumradius.

State the Cosine Rule for a triangle with sides $a, b, c$.

$$c^2 = a^2 + b^2 - 2ab\cos C$$ Rearranged: $\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$.

In coordinate geometry, how do you find the equation of the perpendicular bisector of the segment joining $A$ and $B$?

Find the midpoint $M$ of $AB$, compute the gradient of $AB$, take its negative reciprocal for the perpendicular gradient, then write the line through $M$ with that gradient. Every point on it is equidistant from $A$ and $B$.

How can the perpendicular bisectors of a triangle's sides be used to find a circle through its vertices?

The three perpendicular bisectors meet at the circumcentre, which is equidistant from all three vertices. That distance is the circumradius, giving the circle (circumcircle) passing through all three vertices.

What is the equation of the circle with a given segment from $A(x_1,y_1)$ to $B(x_2,y_2)$ as its diameter?

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$ The centre is the midpoint of $AB$ and the radius is half of $|AB|$.

How do you determine whether a given point lies inside, on, or outside a circle $(x-a)^2 + (y-b)^2 = r^2$?

Evaluate $(x-a)^2 + (y-b)^2$ for the point. If it is $< r^2$ the point is inside; $= r^2$ it is on the circle; $> r^2$ it is outside.

Outline a general strategy for a coordinate-geometry problem combining lines, circles and distances.

Translate the geometry into coordinates/equations; use the distance, midpoint and gradient formulas; apply perpendicularity ($m_1 m_2 = -1$) and tangent/radius conditions; complete the square for circles; and solve simultaneous equations, checking solutions against the original geometric setup.