AAS - Constructing a Tangent Line to a Circle From a Point Outside Circle Lesson
Constructing a Tangent Line to a Circle from a Point Outside Circle
We also need to know how to construct a tangent line to a circle from a point outside the circle.
Steps for Constructing a Tangent to a Circle Through a Point Outside the Circle
- Start with any circle (label the center O) and any point outside the circle (point P).
- Connect point O to point P.
- Construct the perpendicular bisector of segment OP.
- Construct an arc using the midpoint of OP as the center and half the length of OP as the radius.
- Label the points where the arc intersects the circle.
- Construct rays from P through the points of intersection.
- Both rays you have constructed are tangent to the circle!
There is one more important shape we need to talk about. This shape is the cyclic quadrilateral. A cyclic quadrilateral is a quadrilateral inscribed in a circle –that is, all 4 vertices lie on the circle. Cyclic quadrilaterals have some special properties.
The sum of opposite angles of a cyclic quadrilateral is 180 degrees. Let's prove this! We will use the diagram to the left for reference.
To show that the sum of opposite angles of a cyclic quadrilateral is 180 degrees, we need to prove:
Proof
We start by drawing segments DO and BO.
By the Inscribed Angles Theorem, we know that
Similarly, we know
Thus, we know that
We know that m(arcDAB) + m(arcDCB) = 360° since they comprise a whole circle, and there are 360 degrees in a circle.
Then
So,
Since there are in any quadrilateral, then also
Let's try some practice...
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