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4.8 X) Isosceles Triangles in Circles
4.8 X) Isosceles Triangles in Circles
This is not a circle theorem but more a point that comes in handy for answering a variety of different harder circle theorem questions. The point is that when we have a triangle in a circle where one of the points is the centre of the circle and the other two points are on the circumference of the circle, the triangle will be an isosceles triangle. This is because two of the sides in the triangle will be radii, and therefore the same length as each other, thus meaning that the triangle is an isosceles triangle.
There is an example of a triangle in a circle below. The point O is the centre of the circle, and A and B are on the circumference of the circle.
There is an example of a triangle in a circle below. The point O is the centre of the circle, and A and B are on the circumference of the circle.
The sides OA and OB are both radii of the circle, and this means that they are the same length. Therefore, the triangle OAB is an isosceles triangle.
It is useful to know that the triangle is an isosceles triangle because two of the angles in an isosceles triangle are the same size. The two angles that are the same size on the triangle above are the angles that are at the circumference; the angles that are the same size are OAB and OBA. I have labelled these angles as x on the diagram below.
It is useful to know that the triangle is an isosceles triangle because two of the angles in an isosceles triangle are the same size. The two angles that are the same size on the triangle above are the angles that are at the circumference; the angles that are the same size are OAB and OBA. I have labelled these angles as x on the diagram below.
If we were to be given one of the angles in the above triangle, we would be able to work out the sizes of the other angles in the triangle from the isosceles triangle rule, and the rule that the interior angles of a triangle add up to 180°. We are going to have a look at two different examples in this section.
Example 1
The centre of the circle below is O. Point C and D lie on the circumference of the circle. Angle OCD is 50°. Find the sizes of the other two angles.
The centre of the circle below is O. Point C and D lie on the circumference of the circle. Angle OCD is 50°. Find the sizes of the other two angles.
The question tells us that O is the centre of the circle, and C and D are on the circumference of the circle. This means that the lines OC and OD are both radii, thus meaning that triangle OCD is an isosceles triangle. As it is an isosceles triangle, the angles at the circumference will be the same. We are told that angle OCD is 50° and this means that angle ODC is also 50°. I have added this angle onto the diagram.
We now need to find the size of angle COD (which I have labelled as y on the above diagram). We know that all of the interior angles in a triangle add up to 180°. Therefore, we can create the following equation:
We can simplify this equation by collecting the two numbers that are on the left side of the equation (50 and 50). The equation becomes:
We want to find the value of y, which we can do by moving the 100 from the left side of the equation to the right. We are able to do this by taking 100 from both sides of the equation.
This tells us that y is 80°. We now have all of the angles in the triangle; angle ODC is 50° and angle COD is 80°. Both of these angles are shown on the circle below.
Example 2
The centre of the circle below is O. The points E and F lie on the circumference of the circle. Angle EOF is 70°. Find the sizes of the other two angles?
The centre of the circle below is O. The points E and F lie on the circumference of the circle. Angle EOF is 70°. Find the sizes of the other two angles?
The question tells us that O is the centre of the circle, and that E and F are on the circumference of the circle. This means that the lines OE and OF are both radii, thus meaning that the triangle OEF is an isosceles triangle. As OEF is an isosceles triangle, the angles at the circumference will be the same; angles OEF and OFE are the same size. We do not know the sizes of these angles, so I am going to label them as x on the diagram below.
All of the interior angles in a triangle add up to 180°. This means that we can create the following equation.
We can simplify this equation by collect the two x terms on the left side of the equation; the x + x can be collected to become 2x. The equation becomes:
We find the value of x by getting all of the unknowns on one side of the equation and all of the numbers on the other side of the equation. Currently we have 2x on the left and no x’s on the right. Therefore, it makes sense to have the left side of the equation as the side where we get all the x’s to, and the right side of the equation as the side where we get the numbers to. In order to achieve this, we need to move the + 70 from the left side of the equation to the right. We are able to do this by doing the opposite to both sides of the equation. The opposite of adding 70 is taking 70, so we take 70 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.
This tells us that x is 55°; angles OEF and OFE are both 55°. We now have the sizes of all of the interior angles in the triangle and they are all shown on the diagram below.