• Proof That a Triangle Drawn in a Semicircle is Right Angled

    If a triangle is drawn in a semicircle, with the diameter forming one side of the circle, as below, the angle formed opposite the diameter (angle ) is a right angle. The proof follows straight from the fact that the angle subtended by an arc at the...

    https://astarmathsandphysics.com/gcse-maths-notes/632-proof-that-a-triangle-drawn-in-a-semicircle-is-right-angled.html
  • The Myth of the Bermuda Triangle

    Many people have heard of the Bermuda triangle as an area of the Atlantic where ships disappear without trace. Various explanations have been put forward, among them alien abductions, space time vortices – whatever that means – and sea monsters....

    https://astarmathsandphysics.com/gcse-physics-notes/863-the-myth-of-the-bermuda-triangle.html
  • The Myth of the Bermuda Triangle 2

    Many people have heard of the Bermuda triangle where planes or ships disappear without trace. Most famously in 1945, an entire flight of five US Navy TBM Avenger Torpedo bombers disappeared, together with the plane sent to search for them. There are...

    https://astarmathsandphysics.com/gcse-physics-notes/864-the-myth-of-the-bermuda-triangle-2.html
  • Proof of Backwards Form of Triangle Inequality

    The backwards form of the trangle inequality states that for any two vectors and The meaning of the theorem is shown in the following diagram. We can also illustrate with real numbers, which may be considered one dimensional vectors. Take and The proof...

    https://astarmathsandphysics.com/ib-maths-notes/proofs/1080-proof-of-backwards-form-of-triangle-inequality.html
  • Proof of Triangle Inequality

    The meaning of the theorem is obvious from the following diagram. We can also illustrate with real numbers, which may be considered one dimensional vectors. Take and The proof follows from the dot product of with itself. Square rooting both sides now...

    https://astarmathsandphysics.com/ib-maths-notes/proofs/1081-proof-of-triangle-inequality.html
  • Parallelograms and Triangles of Forces

    by measurement or calculation. This method is used most easily for finding the resultant of two forces. We construct a triangle or parallelogram. If a parallelogram is constructed, the resultant is one of the diagonals. Suppose we want to find the...

    https://astarmathsandphysics.com/o-level-physics-notes/200-parallelograms-and-triangles-of-forces.html
  • Trigonometry - Right Angled Triangles - Finding a Side

    There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we have: If we have the angle shown, we label the sides for this angle. Opposite = is the side...

    https://astarmathsandphysics.com/igcse-maths-notes/540-trigonometry-right-angled-triangles-finding-a-side.html
  • The Sine Rule

    If we have a triangle, and we need to calculate angles and/or sides, we can only use simple trigonometry and Pythagoras theorem if the triangle is right angled. If the triangle is not right angled, then we must use general formulae which apply to any...

    https://astarmathsandphysics.com/o-level-maths-notes/361-the-sine-rule.html
  • Angles in Three Dimensional Solids

    Finding the angle between vertices in a three dimensional solid can be tricky. It often helps to construct a triangle between the relevant vertices. Then it is often possible to use trigonometry. Consider the hexagonal pyramid below. The base is a...

    https://astarmathsandphysics.com/gcse-maths-notes/557-angles-in-three-dimensional-solids.html
  • Von Koch or Snowflake Curve

    Von Kck or snowflake curve is the limit curve achieved by the following steps. 1. Start with an equilateral triangle. 2. Divide each side into three equal parts. 3. One the middle third of each side draw an equilateral triangle. 4. Delete the 'base' of...

    https://astarmathsandphysics.com/ib-maths-notes/sequences-and-series/4781-von-kock-or-snowflake-curve.html
  • The Cosine Rule

    If we have a triangle, and we need to calculate angles and/or sides, we can only use simple trigonometry and Pythagoras theorem if the triangle is right angled. If the triangle is not right angled, then we must use general formulae which apply to any...

    https://astarmathsandphysics.com/o-level-maths-notes/357-the-cosine-rule.html
  • Bearings

    Rule: For the above diagram, find a)The distance BC b)The bearing of A from B and the bearing of B from C. a)Label the triangle as above, with sides labelled by littleletters opposite angles labelled by big letters. so b) We draw a right angled triangle...

    https://astarmathsandphysics.com/igcse-maths-notes/449-bearings.html
  • Constructing an Angle of 45 Degrees

    an angle of 45°,start by constructing a right angle. To construct a right angle usethe circle theorem which says that a triangle inscribed in a circle,one edge of which is a diameter, is always a right angled triangle. Therefore, start by drawing a...

    https://astarmathsandphysics.com/igcse-maths-notes/457-constructing-an-angle-of-45-degrees.html
  • Bearings

    Rule: For the above diagram, find a)The distance BC b)The bearing of A from B and the bearing of B from C. a)Label the triangle as above, with sides labelled by little letters opposite angles labelled by big letters. so b) We draw a right angled...

    https://astarmathsandphysics.com/gcse-maths-notes/560-bearings.html
  • Constructing an Angle of 45 Degrees

    angle of 45 ° , start by constructing a right angle. To construct a right angle use the circle theorem which says that a triangle inscribed in a circle, one edge of which is a diameter, is always a right angled triangle. Therefore, start by drawing a...

    https://astarmathsandphysics.com/gcse-maths-notes/571-constructing-an-angle-of-45-degrees.html
  • The Tangent Function

    The tangent function is one of the three important trigonometric function. For an angle in a right angled triangle is the the ratio of the side opposite the angle and the side adjacent to the angle. For right angled triangles and We can write If we...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1147-the-tangent-function.html
  • Angles in Polygons

    Shape Picture Number of Sides Interior Angle Sum of Interior Angles Exterior Angle Triangle 3 60 180 120 Square 4 90 360 90 Pentagon 5 108 540 72 Hexagon 6 120 720 60 Octagon 8 135 1080 45 n- agon To see why the angles in a triangle add to notice that...

    https://astarmathsandphysics.com/o-level-maths-notes/318-angles-in-polygons.html
  • Quadratics

    deriving equations to represent problems and solving them. A very good example is found from geometry. The right angled triangle below has three unknown sides. Believe it or not, we can find the lengths of the sides knowing only the relationships shown...

    https://astarmathsandphysics.com/ib-maths-notes/quadratics/1086-quadratics.html
  • The Tetrahedron

    The tetrahedron is a four sided shape. Each side is a equilateral triangle. We can find angles in the tetrahedron using only the properties of right angled triangles – simple triogonometry and Pythagoras theorem. If the length of each side is 1, the...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1148-the-tetrahedron.html
  • Proof Length of Tangent Squared Equals Product of Chord With Chord Extended to Meet Tangent

    squared equals product of chord with chord extended to meet the tangent. Proof Draw chords AD and BD. Now consider the triangles and Angle is common to both triangles, so call it Call angle then other angles can be found in terms of and Now draw radii...

    https://astarmathsandphysics.com/igcse-maths-notes/504-proof-length-of-tangent-squared-equals-product-of-chord-with-chord-extended-to-meet-tangent.html

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