Proving Triangles are Congruent

To prove two triangles are congruent, it is sufficient two any of the following: The lengths of corresponding sides of each triangle are the same. If triangle 1 has lengths 4, 5 and 6 cm, and triangle 2 has lengths of 4, 5 and 6 cm, then the triangles are congruent. Each triangle has two […]

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Vectors II

Vectors can be added in an obvious way. In the diagram belowgoes fromto B and goes fromto The vectorgoes fromtoWe can writeor .Importantly, so Ifis split in some ratio by some pointthen we can findusing this ratio. Suppose splitsin the ratio 2:3, thenis in the ratio 2/3 so that and We can label the triangle as below. Then  For a slightly […]

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Transformations of Graphs

We can sort transformations of graphs into two types -x transformations or y transformations. Anything else is a combination of an x transformation followed by a y transformation or vice versa. A transformation is an x transformation if it is an argument of a function on the right hand side, or if it can be […]

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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 regular pyramid and all the sides are the same length, 2cm. The pyramid is […]

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Angles of Elevation and Depression

The angle of elevation of an object is the angle a line of sight to the object makes above the horizontal, and the angle of depression is the angle of a line of sight to the object below the horizontal. We can find angles of elevation or depression using simple trigonometry. Consider the diagram below. […]

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Bisecting an Angle

We can bisect an angle using only compasses. Put the point of the compass at O and draw arcs that crosses both lines that constitute the angle. From each point where an arcs just drawn cross a line construct other arcs to cross as shown. Now draw a line from O to where the two […]

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