Tangents - Higher
There are two circle A mathematical statement that can be demonstrated to be true. involving A straight line that just touches a point on a curve. A tangent to a circle is perpendicular to the radius which meets the tangent..
1. The angle between a tangent and a The distance from the centre of a circle to its circumference. The plural of radius is radii. is 90°.
2. Tangents which meet at the same point are equal in length.
Example
Calculate the angles EFG and FOG.
Triangle GEF is an Two sides have equal lengths. Angles opposite the equal sides are equal. triangle.
Angle FGE = angle EFG
FGE = EFG = \(\frac{180 - 20}{2} = 80^\circ\)
The angle between the tangent and the radius is 90°.
Angle EFO = EGO = 90°
The shape FOGE is a A quadrilateral is a shape with four straight sides and four angles.. The angles in a quadrilateral add up to 360°.
Angle FOG = \(360 - 90 - 90 - 20 = 160^\circ\)
Proof
The angle between the tangent and the radius is 90°.
Angle BCO = angle BAO = 90°
AO and OC are both radii of the circle.
Length AO = Length OC
Draw the line OB. It creates two triangles OCB and OAB. These share the length OB.
Triangles OCB and OAB are Congruent means two shapes are exactly the same shape and exactly the same size. because of the SAS rule.
Two of the sides are the same length: OB = OB and OC = OA
One of the angles is equal in size: OCB = OAB
Congruent triangles are identical.
So length CB = AB.