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4.8 F) Tangents – Part 2
4.8 F) Tangents – Part 2
We learnt in the previous section that a tangent is a line that just touches a curve. There is a circle with a tangent drawn below.
There are two circle theorems that involve tangents. One of these circle theorems is that the angle between a tangent and a radius is 90°; we looked at this circle theorem in the previous section (click here to be taken to the previous section).
The second circle theorem to do with tangents is that two tangents to a circle that meet at the same point are equal in length. They are equal in length from the point where they are a tangent to the point where the two tangents intersect one another.
The second circle theorem to do with tangents is that two tangents to a circle that meet at the same point are equal in length. They are equal in length from the point where they are a tangent to the point where the two tangents intersect one another.
There are two tangents on the above diagram. One of the tangents in the line ABC, which touches the circle at point B. The other tangent is the line DEC, which touches the circle at point E. The two tangents meet at the point C. The circle theorem states that two tangents that meet at the same point will be equal in length. This means that BC and EC will be equal in length.
Example 1
The straight line ABC is a tangent to the circle at B and the straight line ADE is a tangent to the circle at D. Find the size of angle ABD and OBD. Give reasons for your answers.
The straight line ABC is a tangent to the circle at B and the straight line ADE is a tangent to the circle at D. Find the size of angle ABD and OBD. Give reasons for your answers.
We can see from the diagram that the two tangents meet each other at point A. The circle theorem that we have just looked at states that two tangents that meet at the same point are equal in length. This means that AB and AD are equal in length. We can show that these two lines are the same length by adding a small dash on the lines. I have added these dashes to the diagram below. I have also labelled the two angles that we are looking for; I have labelled angle ABD as x and angle OBD as y.
Let’s find the size of angle x first and we are going to find the size of angle x by looking at the triangle ABD. From the above working, we know that AB and AD are the same length, which means that the triangle ABD is an isosceles triangle. We can now say that the angle ABD and angle BDA are equal to one another. Therefore, we can say that both of these angles are x.
We are now in a position to work out the size of angle x in the triangle. We know that all of the angles inside a triangle add up to 180° and from this information, we can create the following equation:
The first step in finding the value of x is to simplify the left side of the equation; we can collect the two x terms.
We want to isolate the x terms; we want all of the x terms on one side of the equation and all of the numbers on the other side of the equation. I am going to get all of the x terms on the left side of the equation and all of the numbers on the right. Therefore, we need to move the 30 that is currently on the left to the right, which we are able to do by taking 30 from both sides of the equation.
We want to find the value of x and not 2x. Therefore, we need to divide both sides of the equation by 2.
We can now say that angle ABD is 75°.
The second part of the question asks us to find the size of angle OBD (which is the angle labelled y on the diagram below).
The second part of the question asks us to find the size of angle OBD (which is the angle labelled y on the diagram below).
We are able to find the size of this angle by using the other circle theorem to do with tangents. This circle theorem states that a tangent meets a radius at 90°. This means that angle ABO is 90°. Angle ABO is made up of two smaller angles; angle ABD and OBD. The sum of angle ABD and OBD will be 90°. We found in the first part of the question that angle ABD is 75°. We can create the following equation from this information.
We find the value of y by moving the 75 from the left side of the equation to the right, which we can do by taking 75 from both sides of the equation.
Therefore, angle OBD is 15°.