e $OQ$ and $OR$ are both radii of the circle, they are equal in length ($OQ = OR$). This makes $\\triangle OQR$ an isosceles triangle.\n\nSince base angles in an isosceles triangle are equal:\n\n$$\\angle OQR = \\angle ORQ = 36^\\circ$$"},{"type":"text","order":2,"title":"Calculate angle QOR","content":"The angles in any triangle add up to $180^\\circ$. Using this for $\\triangle OQR$:\n\n$$\\begin{aligned} \\angle QOR &= 180^\\circ - \\angle ORQ - \\angle OQR \\\\ \\angle QOR &= 180^\\circ - 36^\\circ - 36^\\circ \\\\ \\angle QOR &= 108^\\circ \\end{aligned}$$"},{"type":"text","order":3,"title":"Use congruent triangles","content":"Now consider triangles $\\triangle OPQ$ and $\\triangle ORQ$:\n- $OP = OR$ (both are radii of the circle)\n- $OQ$ is a shared side\n- $PQ = QR$ (given in the question)\n\nBy SSS (Side-Side-Side), the two triangles are congruent:\n\n$$\\triangle OPQ \\cong \\triangle ORQ$$\n\nBecause they are congruent, their corresponding angles are equal, meaning:\n\n$$\\angle POQ = \\angle QOR = 108^\\circ$$","highlights":[{"text":"$$PQ = QR$","location":"specification"}]},{"type":"text","order":4,"title":"Calculate the final angle","content":"We want to find the total angle $\\angle POR$. Looking at the diagram, we can see that:\n\n$$\\angle POR = \\angle POQ + \\angle QOR$$\n\nSubstituting our known values:\n\n$$\\angle POR = 108^\\circ + 108^\\circ = 216^\\circ$$\n\nSince this refers to the reflex angle, the interior angle $\\angle POR$ is:\n\n$$\\angle POR = 360^\\circ - 216^\\circ = 144^\\circ$$"}]}],"generatedAt":"2026-06-12T13:08:12.608Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":21}},{"id":"2f7320af-c4a2-4ac8-a22e-04e5a16c0e8a","revisionId":"f105dced-2399-4afb-83ee-40c2944c1ab6","questionNumber":null,"questionType":"LA","specification":"![A geometric diagram showing a circle with center O. A horizontal line SMT is tangent to the circle at point M. Points K and L are on the circle. Line segments MK, ML, and OL are drawn. A line segment KL is also drawn. There are single tick marks on segments KL and LM, indicating they are equal in length. The angle SMK is labeled as y degrees.](https://assets.mathsgenie.co.uk/questions/7fb22231-c179-4e87-98f0-a06999b99b5a.png)\n\n$K$, $L$ and $M$ are points on a circle, centre $O$.\n$SMT$ is a tangent to the circle at $M$.\n\n$KL = LM$\nAngle $SMK = y^\\circ$","difficulty":4,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"95c092b2-80e9-444f-be6d-720add79fa67","label":"a","content":"Find the size of angle $OML$, in terms of $y$.\nGive reasons for each stage of your working.","markscheme":"- $\\angle KLM = y$ (Alternate segment theorem) M1\n- $\\angle KML = (180 - y) / 2 = 90 - \\frac{1}{2}y$ (Angles at the base of an isosceles triangle are equal) M1\n- $\\angle OMS = 90^\\circ$ (Tangent meets radius at $90^\\circ$) M1\n- $\\angle OMK = 90 - y$ M1\n- $\\angle OML = (90 - \\frac{1}{2}y) - (90 - y) = \\frac{1}{2}y$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"States $\\angle KLM = y$ with reason referencing Alternate Segment Theorem.","code":"M"},{"id":"p2","marks":1,"description":"Calculates $\\angle KML = 90 - \\frac{1}{2}y$ with reason referencing isosceles triangle properties.","code":"M"},{"id":"p3","marks":1,"description":"States $\\angle OMS = 90^\\circ$ with reason referencing tangent-radius property.","code":"M"},{"id":"p4","marks":1,"description":"Correct method to find $\\angle OMK$, e.g., $90 - y$.","code":"M"},{"id":"p5","marks":1,"description":"Final simplified expression for $\\angle OML$.","code":"A"}],"finalAnswers":["\\frac{1}{2}y","0.5y","y/2"],"generatedAt":"2026-05-06T14:07:20.809Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"angle_chasing","label":"Angle chasing with circle theorems","steps":[{"type":"text","order":0,"title":"Use the tangent theorem","content":"The angle between tang

Circle Theorems

Question 1

Circle Theorems Questions

Circle Theorems

Question 1

Circle Theorems Questions