Geometry

The One Thing

Geometry problems in this material are solved by identifying the governing relationship: proportionality in similar figures, a standard locus condition, a circle theorem, or a precise construction procedure. Accurate diagrams, correct labels, and explicit use of the relevant condition are essential.

Definitions and Results

• Similar triangles: Triangles with equal corresponding angles and proportional corresponding sides. If triangles $ABC$ and $DEF$ are similar, then

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}.$$

• AA similarity criterion: Two triangles are similar if two corresponding angles of one triangle are equal to two corresponding angles of the other.
• SAS similarity criterion: Two triangles are similar when one pair of corresponding angles is equal and the including sides are proportional.
• SSS similarity criterion: Two triangles are similar when all three pairs of corresponding sides are proportional.
• Scale factor: The ratio of any corresponding length in one similar figure to the matching length in the other. If the side ratio is $k$, corresponding perimeters are in the ratio $k$, and corresponding areas are in the ratio $k^2$.
• Basic proportionality theorem: If $DE$ is parallel to $BC$ in triangle $ABC$, then

$$\frac{AD}{DB}=\frac{AE}{EC}.$$

• Converse of the basic proportionality theorem: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
• Pythagoras theorem: In a right-angled triangle,

$${\text{hypotenuse}}^2={\text{perpendicular}}^2+{\text{base}}^2.$$ The triangle must be right-angled.

• Converse of Pythagoras theorem: If the square of the longest side equals the sum of the squares of the other two sides, the triangle is right-angled.
• Locus: The set or path of all points satisfying a specified condition.
• Perpendicular bisector locus: The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.
• Angle-bisector locus: The locus of points equidistant from the two arms of an angle is its angle bisector.
• Fixed-distance locus from a point: The locus of points at a fixed distance from a fixed point is a circle with that point as centre.
• Fixed-distance locus from a line: The locus of points at a fixed distance from a fixed line consists of two lines parallel to the given line, one on each side at that distance.
• Circle: The set of all points in a plane at a fixed distance from a fixed point called the centre.
• Chord: A line segment joining any two points on a circle.
• Arc: A part of the circumference of a circle between two points.
• Tangent: A line that touches a circle at exactly one point.
• Cyclic quadrilateral: A quadrilateral whose four vertices lie on the same circle.
• Concyclic points: Points that lie on one common circle.
• Circle-chord theorem: The perpendicular from the centre of a circle to a chord bisects the chord.
• Equal-chord theorem: Equal chords of a circle are equidistant from the centre, and chords equidistant from the centre are equal.
• Angle at the centre: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circle.
• Angles in the same segment: Angles in the same segment of a circle are equal.
• Angle in a semicircle: The angle in a semicircle is a right angle.
• Cyclic quadrilateral angle theorem: Opposite angles are supplementary:

$$\angle A+\angle C=180^{\circ },\angle B+\angle D=180^{\circ }.$$

• Exterior angle theorem for a cyclic quadrilateral: The exterior angle equals the interior opposite angle.
• Tangent-radius theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
• Equal tangents theorem: Tangents drawn from an external point to a circle are equal in length.
• Alternate segment theorem: The angle between a tangent and a chord equals the angle in the alternate segment.
• Construction: The accurate drawing of geometrical figures using instruments such as a ruler, compass, and protractor where permitted.
• Triangle construction condition: A triangle can be constructed from three given sides only when the sum of any two sides is greater than the third side.

Worked Methods

Using similarity to calculate unknown lengths

• Draw a careful figure and identify corresponding vertices, sides, and angles.
• Establish similarity using AA, SAS, or SSS.
• Match corresponding sides consistently.
• Write the proportional relationship. For triangles $ABC$ and $DEF$,

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}.$$

• Solve the resulting equation for the unknown length.
• If required, use the scale factor $k$: perimeters are in ratio $k$, and areas are in ratio $k^2$.

Applying the basic proportionality theorem

• In triangle $ABC$, identify points $D$ and $E$ on two sides.
• Confirm that $DE\parallel BC$.
• Use

$$\frac{AD}{DB}=\frac{AE}{EC}.$$

• Substitute the known lengths and solve for the unknown.

Using the converse of the basic proportionality theorem

• Identify a line dividing two sides of a triangle.
• Calculate the two side-division ratios.
• If the ratios are equal, apply the converse:

$$\frac{AD}{DB}=\frac{AE}{EC}\Rightarrow DE\parallel BC.$$

• State the parallel-line conclusion explicitly.

Applying Pythagoras theorem

• Confirm that the triangle is right-angled.
• Identify the hypotenuse as the side opposite the right angle.
• Use

$${\text{hypotenuse}}^2={\text{perpendicular}}^2+{\text{base}}^2.$$

• Substitute the known lengths and solve.
• For a proof that a triangle is right-angled, use the converse of Pythagoras theorem by checking whether the square of the longest side equals the sum of the squares of the other two sides.

Constructing a perpendicular bisector

• Draw the segment whose perpendicular bisector is required.
• Using a compass, draw equal arcs from the two endpoints of the segment.
• Ensure that the arcs intersect on both sides of the segment.
• Join the two arc intersections with a straight line.
• This line is the perpendicular bisector because every point on it is equidistant from the two endpoints.

Constructing an angle bisector

• From the vertex, draw an arc cutting both arms of the angle.
• From the two points where the arc cuts the arms, draw equal arcs.
• Mark their new intersection inside the angle.
• Join the vertex to this new intersection.
• The resulting line is the angle bisector because its points are equidistant from the two arms.

Constructing a tangent at a point on a circle

• Identify the point of contact on the circle.
• Join that point to the centre of the circle.
• Draw a perpendicular to the radius at the point of contact.
• The perpendicular line is the tangent, since the tangent at any point is perpendicular to the radius through that point.

Constructing tangents from an external point

• Let the external point be joined to the centre of the circle.
• Bisect the segment joining the external point to the centre.
• Draw a circle with the midpoint as centre and with radius equal to the distance from the midpoint to either endpoint.
• Use the intersections of this circle with the given circle to obtain the points of contact.
• Join the external point to the points of contact.
• The resulting lines are the tangents; the two tangent lengths are equal.

Constructing a triangle from three sides

• Check that the sum of any two given sides is greater than the third side.
• Draw one given side with a ruler.
• With the compass set to the length of a second side, draw an arc from one endpoint.
• With the compass set to the length of the third side, draw an arc from the other endpoint.
• Use the arc intersection as the third vertex.
• Join the third vertex to both endpoints and label the triangle.

Identifying a locus

• State the condition defining the moving point.
• Classify it as a standard condition:

- equal distance from two points; - equal distance from two lines; - fixed distance from a point; - fixed distance from a line.

• Construct or describe the corresponding locus:

- perpendicular bisector; - angle bisector; - circle; - two parallel lines.

• Check that every point on the proposed locus satisfies the original condition.

Applying circle theorems

• Draw and label the circle accurately.
• Identify the relevant chord, radius, tangent, arc, or cyclic quadrilateral.
• Select the applicable theorem:

- perpendicular from centre to chord; - equal chords and equal distances from centre; - angle at the centre; - angles in the same segment; - angle in a semicircle; - opposite angles of a cyclic quadrilateral; - exterior angle of a cyclic quadrilateral; - tangent-radius theorem; - equal tangents theorem; - alternate segment theorem.

• Write the angle or length relationship.
• Substitute known values and solve.

Where It Goes Wrong

• Corresponding sides in similar triangles are matched inconsistently; the correspondence must follow the equal corresponding angles.
• The basic proportionality theorem is used without confirming that $DE\parallel BC$, or its converse is applied without first showing equal side-division ratios.
• Pythagoras theorem is applied when the triangle has not been established as right-angled, or the hypotenuse is misidentified.
• A locus is confused with a single point or one line: fixed distance from a point gives a circle, while fixed distance from a line gives two parallel lines.
• Circle theorems are applied without identifying the relevant chord, radius, tangent, arc, or cyclic quadrilateral.
• A construction omits a required condition, uses unequal arcs where equal arcs are required, or fails to check the triangle inequality for three given sides.

What Gets Asked

• Prove or identify similar triangles using AA, SAS, or SSS.
• Calculate unknown lengths using corresponding-side ratios and scale factors.
• Use the basic proportionality theorem or its converse to establish a ratio or prove parallel lines.
• Apply Pythagoras theorem to calculate a side, or use its converse to prove that a triangle is right-angled.
• Describe or construct loci involving equal distances from two points, equal distances from two lines, or fixed distances from a point or line.
• Apply circle theorems involving chords, arcs, angles at the centre, angles in the same segment, semicircles, tangents, and cyclic quadrilaterals.
• Prove that points are concyclic or determine unknown angles in a cyclic quadrilateral.
• Construct perpendicular bisectors, angle bisectors, triangles from three sides, tangents at a point, and tangents from an external point.
• Explain why a construction is valid and verify that it satisfies every given condition.