Back to Edexcel Circle Theorems (H) Home
4.8 E) Tangents – Part 1
4.8 E) Tangents – Part 1
A tangent is a straight line that just touches a curve at a particular point. There is a circle with a tangent drawn below.
The tangent on the diagram above touches the circle at point A.
There are two circle theorems to do with tangents and circles. The first circle theorem to do with tangents is that the angle between a tangent a radius is 90° (a right angle).
There are two circle theorems to do with tangents and circles. The first circle theorem to do with tangents is that the angle between a tangent a radius is 90° (a right angle).
The angle OAB and OAC are both 90°.
Example 1
The straight line ECD is a tangent to the circle at point C. Find the size of the angle BCD.
The straight line ECD is a tangent to the circle at point C. Find the size of the angle BCD.
We are told in the question that ECD is a tangent to the circle at point C. We can also see from the diagram that the line OC is a radius for the circle. This means that the angle ECO and the angle DCO will both be 90°; the circle theorem states that the angle between a tangent and a radius is 90° (a right angle).
We are trying to find the size of angle BCD. This angle and the angle that is 54° (angle BCO) will add up to 90° because angle DCO is 90°. We are able to create the following equation from this information (I will let angle BCD equal y):
We are trying to find the size of angle BCD. This angle and the angle that is 54° (angle BCO) will add up to 90° because angle DCO is 90°. We are able to create the following equation from this information (I will let angle BCD equal y):
We find the value of y by moving the 54 from the left side of the equation to the right, which we are able to do by doing the opposite; we take 54 from both sides of the equation.
Therefore, y is 36°.
Example 2
The straight line ABC is a tangent to the circle at point B. The straight line EFC is a tangent to the circle at point F. What is the size of angle BOF?
The straight line ABC is a tangent to the circle at point B. The straight line EFC is a tangent to the circle at point F. What is the size of angle BOF?
We are given two lines that are tangents to the circle in the question (the lines ABC and EFC are tangents to the circle). We are also given two radii for the circle; BO and FO are both radii. The circle theorem states that the angle between a tangent and a radius is 90°. Therefore, angle CBO and angle CFO are 90°. We can add this information to the diagram. I have also labelled the angle that we are looking for as x (the angle that we are looking for is BOF).
The next step in answering this question is to look at the quadrilateral BCFO. We have 3 of the 4 angles in this quadrilateral and the angle that we do not know is the angle that we are trying to find out. We know that the angles inside a quadrilateral add up to 360° and from this information, we can create the following equation:
We now collect the numbers on the left side of the equation.
We want to find the value of x, which we are able to do by moving the 227 from the left side of the equation to the right. We are able to move the 227 from the left to the right by taking 227 from both sides of the equation.
Therefore, x is 133°.