Archive | Geometry

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 […]

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 […]

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 […]

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 […]

Proof That Angles Subtended in a Circle From the Same Chord (Angles in the Same Segment) Are Equal

The indicated anglesin the diagram below are equal. By drawing in the chord AB (below) the anglescan be seen to be in the same segment of the circle and subtended from the same chord. The equality of the angles is then a consequence of the fact that(1) This last is another circle theorem, proved here.

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 centre of a circle is twice the angle subtended […]

Area and Perimeter of a Sector (Part of a Circle)

The area of a circle isand the perimeter or circumference isA sector is a fraction of a circle. To find the area of a sector, we just find that fraction of the area of a circle. The sector to the right is a fractionof the circle to the left so the the area of the […]

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. […]

Finding the Locus of Points Equidistant From Two Given Points

Intuitively the set of points equidistant from two given points is the set of points halfway between them. The locus will be a straight line halfway between the two points. We can construct this line by drawing circles of equal radius centred at A and B. The radius should be over half the distance AB. […]

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 […]