GCSE Higher Maths: Algebraic and Geometric Proof (Non-Calculator)
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GCSE Higher Maths: Algebraic and Geometric Proof (Non-Calculator): 6 questions, marked instantly with every answer explained. Free, no sign-up, no adverts in the test. This test is for GCSE (Years 10 and 11, ages 14–16). In the US, that's about Grades 9 and 10.
Before you start
6 questions on identities, counterexamples and showing that a statement is always true, picked at random from a bank of 9, so every try is different. Non-calculator: work these out without a calculator. Type a number, or choose an answer. Questions with units tell you which unit to use. Choose Practice to see each answer explained as you go, or Test to see your score and explanations at the end.
- 6 questions – a new mix each time you try
- Plus written questions for your teacher or a parent to check
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Written questions – check with your teacher
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XYZD is a parallelogram. Its diagonals cross at E. Prove that E is the midpoint of both diagonals.
4 marks
Marking guide (for your teacher or a parent)
Give a mark for each point the answer shows. Tick as you go:
One answer that would get full marks:
XY is parallel to DZ, so angle EXY = angle EZD and angle EYX = angle EDZ (alternate angles). XY = DZ (opposite sides of a parallelogram). So triangles XEY and ZED are congruent (ASA). So XE = EZ and YE = ED: the diagonals bisect each other.
OA = a and OB = b. M is the midpoint of AB. P is the point with OP = 4/2(a + b). Show that OP = 4OM and explain what this tells you about O, M and P.
4 marks
Marking guide (for your teacher or a parent)
Give a mark for each point the answer shows. Tick as you go:
One answer that would get full marks:
AB = b − a, so OM = OA + ½AB = a + ½(b − a) = ½(a + b). OP = 4/2(a + b) = 4 × ½(a + b) = 4OM. OP is a multiple of OM and both start at O, so O, M and P are on one straight line, with OP 4 times as long as OM.
Triangle XYZ has XY = XZ. M is the midpoint of YZ. Prove that angle XYZ = angle XZY.
4 marks
Marking guide (for your teacher or a parent)
Give a mark for each point the answer shows. Tick as you go:
One answer that would get full marks:
In triangles XYM and XZM: XY = XZ (given), YM = ZM (M is the midpoint of YZ) and XM is a side of both. So the triangles are congruent (SSS). Matching angles of congruent triangles are equal, so angle XYZ = angle XZY.
GCSE-style practice by Classfern. Not an official test and not affiliated with or endorsed by any exam board, Ofqual or the Department for Education.
Curriculum: the GCSE subject content covered here – A6 – know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs.
Contains public sector information licensed under the Open Government Licence v3.0. Curriculum statements from: Department for Education, Mathematics GCSE subject content and assessment objectives, © Crown copyright 2013.
Learn it first: Algebraic proof and identities (Year 11)
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