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Geometry in the Indian Olympiad Qualifier in Mathematics (IOQM) looks harder than it is. The questions rarely need advanced theorems. They need the Class 9 and 10 facts used with confidence, an accurate figure, and the habit of looking for similar triangles. This post solves three geometry questions from the real IOQM 2024 paper and lists the facts each one used.
What geometry does IOQM expect?
The Syllabus for Mathematical Olympiad on the IOQM site says aspirants must know all topics in the NCERT mathematics books for Classes 8, 9 and 10, and lists for plane geometry "triangles, quadrilaterals, circles and their properties", concurrency and collinearity (Ceva and Menelaus), sine and cosine rules, and properties of triangles and polygons. The IOQM 2024 paper had 30 questions with integer answers, and its official answer key gives 2, 3 or 5 points per question. Check the current year's pages for the format.
Question 4: angle chasing in a quadrilateral
ABCD is a quadrilateral with ∠ADC = 70°, ∠ACD = 70°, ∠ACB = 10° and ∠BAD = 110°. Find ∠CAB.
In triangle ACD, the angles at D and C are both 70°, so ∠CAD = 180° − 140° = 40°.
The diagonal AC splits ∠BAD, so ∠CAB = ∠BAD − ∠CAD = 110° − 40° = 70°.
The official answer is 70. The value ∠ACB = 10° is not needed for this answer. Olympiad questions sometimes give more than you need, so do not assume every number must be used.
Question 12: similar triangles from parallel lines
Square ABCD has side 16. Points E and F on CD satisfy CE = EF = FD. Lines BF and AE meet at M. Find the area of triangle MAB.
EF = 16/3, and EF is parallel to AB.
M is where AE and BF cross, so triangles MAB and MEF are similar, with A matching E and B matching F.
The ratio of similarity is AB/EF = 16 ÷ 16/3 = 3. Their heights from M are in the same ratio, and together they make the side of the square: h + h/3 = 16, so h = 12.
Area of MAB = ½ × 16 × 12 = 96.
Check it with coordinates. Put A at (0, 0) and B at (16, 0), so E = (32/3, 16) and F = (16/3, 16). Line AE is y = 1.5x. Line BF meets it at height 12, which agrees. The official answer is 96.
Question 17: a chord in the circumcircle
Triangle ABC has BC = 30 and AB = AC = 20. D is the foot of the perpendicular from A to BC, M is the midpoint of AD, and PQ is a chord of the circumcircle through M, parallel to BC. Find PQ.
BD = 15, so AD = √(20² − 15²) = √175 = 5√7.
Area K = ½ × 30 × 5√7 = 75√7.
Circumradius R = abc/(4K) = (20 × 20 × 30)/(300√7) = 40/√7.
The centre O lies on line AD at distance R from A. AM = AD/2 = 17.5/√7, so OM = R − AM = 22.5/√7.
PQ is perpendicular to AD, and the perpendicular from the centre bisects a chord, so (PQ/2)² = R² − OM² = (1600 − 506.25)/7 = 156.25. PQ/2 = 12.5, so PQ = 25.
The official answer is 25. Notice that R = 40/√7 is more than AD = 35/√7, so O lies below BC. That fits, since cos A = (400 + 400 − 900)/800 is negative and the angle at A is obtuse. Only a careful figure shows this.
Why does drawing an accurate figure help so much?
Larkin and Simon (1987) compared diagrams with written descriptions that hold the same information. In a diagram, information is organised by location, and facts that are only implicit in text become explicit, so the next step of an inference is often right next to the current one. For olympiad geometry, that means a figure drawn to scale with ruler and compass will often show you the similar triangles or the obtuse angle before any algebra does.
Common mistakes and how to avoid them
Sketching a "generic" triangle when the data say isosceles or obtuse. Use the given lengths.
Mixing up which vertices match in similar triangles. Write the correspondence, such as A to E and B to F, before writing ratios.
Forgetting the circumcentre can lie outside the triangle. Check whether the triangle is acute or obtuse first.
Not checking with a second method. Coordinates are slow but reliable for confirming a synthetic answer.
The short list of facts to know well
Angle sum in a triangle, exterior angle, and isosceles base angles.
Similar triangles (AA), and that areas scale as the square of the ratio.
Pythagoras, and area as ½ × base × height.
Circle facts: equal chords, angle in a semicircle, angles in the same segment, cyclic quadrilaterals, perpendicular from the centre bisects a chord.
R = abc/(4K), and the sine rule a/sin A = 2R.
Our post on similar triangles for Class 10 revises the first group.
A practice routine for IOQM geometry
Take the geometry questions from two past papers on the HBCSE past papers page.
For each, draw a figure to scale before reading any solution.
Solve it once by synthetic geometry and once by coordinates or trigonometry, and compare.
Write the fact that cracked it on a card. After ten questions, your cards are your revision list.
If you get stuck on why a step works, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Next step: redo Question 12 with CE = EF = FD replaced by CE = FD = 4, and check your answer against a coordinate calculation.
