Playbook
Triangles
151 q · 20% HARD · about seven a paper. The altitude to the hypotenuse is 27 questions with no HARD at all; triangle inequalities and the sine and cosine rules are half HARD but only 16 questions between them.
- Questions in the bank
- 151
- q/paper (21 papers)
- 7.19
- Tagged HARD
- 20%
- Subtopics
- 10
Strand: Cornerstone
When you’ll see it
A triangle with sides, angles, medians, altitudes or similar parts given.
How this chapter is tested
151 questions, about seven a paper and steady. The altitude to the hypotenuse is 27 questions with no HARD at all — the single cheapest page in the geometry block.
The expensive pages are small: triangle inequalities and the sine and cosine rules are about half HARD but hold only 16 questions between them; medians and Apollonius (43%) and the centres of a triangle (31%) cost more per question than the rest.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Angles and the exterior angle
Angle sum 180°; an exterior angle equals the two opposite interior angles.
Pythagoras and its converse
Test a² + b² against c² to classify a triangle as acute, right or obtuse.
The altitude to the hypotenuse
h² = pq and h = ab/c for the altitude on the hypotenuse.
Similarity and areas
Areas of similar triangles are in the ratio of the squares of their sides.
Medians and centres
Apollonius: AB² + AC² = 2(AD² + BD²); the centroid divides a median 2 : 1.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.
Area ratio as side ratio
Similar triangles with sides 2 : 3 have areas 4 : 9.
Triangle inequality at equality
Sides 3, 4, 7 make no triangle — the sum must EXCEED the third side.
Centroid and circumcentre
Only in an equilateral triangle do the centres coincide.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each subtopic. Read the notes once, then drill subtopic by subtopic below.
Triangles notesDrill every Triangles question
151 questions from the bank, across 10 subtopics.
Drill one subtopic at a time
The 10 subtopics, in teaching order.
- The Altitude to the HypotenuseDrill The Altitude to the Hypotenuse
- Parallels, Midpoints and the Bisector TheoremDrill Parallels, Midpoints and the Bisector Theorem
- Angles of a TriangleDrill Angles of a Triangle
- Ratio of Areas of TrianglesDrill Ratio of Areas of Triangles
- Pythagoras Theorem and its ConverseDrill Pythagoras Theorem and its Converse
- Congruence and SimilarityDrill Congruence and Similarity
- Centres of a TriangleDrill Centres of a Triangle
- Medians and Apollonius TheoremDrill Medians and Apollonius Theorem
- Triangle InequalitiesDrill Triangle Inequalities
- Sine and Cosine RulesDrill Sine and Cosine Rules
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