QR is tangent at Q to circle whose centre is P PR it parallel to AQ where AQ is chord through A end point of diameter AB prove that BR is tangent at B

To tackle this problem, let's visualize the scenario with the circle and the tangents. We have a circle with center P, and there is a diameter AB. A chord AQ is drawn through point A, which is one end of the diameter. The line QR is tangent to the circle at point Q, and we know that PR is parallel to AQ. Our goal is to prove that line BR is tangent to the circle at point B.

Understanding the Geometry of the Circle

First, let's break down what we know about circles and tangents. A tangent to a circle is a line that touches the circle at exactly one point. The fundamental property of a tangent line is that it is perpendicular to the radius at the point of tangency. So, if BR is to be tangent to the circle at point B, we need to show that it satisfies this condition.

Identifying Key Elements

• Circle: Center P, diameter AB.
• Chord: AQ, where A is one endpoint of the diameter.
• Tangent: QR, which touches the circle at point Q.
• Line: PR is parallel to AQ.

Analyzing the Relationships

Since PR is parallel to AQ, we can infer several geometric relationships. When two lines are parallel, corresponding angles formed by a transversal are equal. In our case, if we draw line BR, which we want to prove is a tangent at B, we must consider the angles formed by the radius PB and line BR.

Because AB is a diameter, we know that angle APB is a right angle (90 degrees). Since QR is tangent at Q, the angle between the radius PQ and line QR is also 90 degrees. This relationship is crucial because it gives us a pathway to demonstrate that BR must also be perpendicular to radius PB.

Establishing the Tangent Condition

To prove that BR is tangent at B, we need to show that angle PBR is 90 degrees. Since PR is parallel to AQ, and AQ is a chord, we can extend this reasoning:

• Since PR is parallel to AQ, angle PAR (formed between AQ and line BR) is equal to angle PQR (formed between QR and line PB) because they are corresponding angles.
• Since angle APB is 90 degrees, it follows that angle PQR must also be 90 degrees because it is the angle between the radius and the tangent line QR.
• Thus, angle PBR must also be 90 degrees, as it is vertically opposite to angle PQR.

Conclusion of the Argument

With angle PBR established as 90 degrees, we confirm that line BR is indeed perpendicular to radius PB. Therefore, by definition, BR is tangent to the circle at point B. This completes our proof that BR is tangent at B, based on the relationships of the angles and the properties of tangents and circles.

In summary, through careful analysis of the geometric properties and relationships, we have demonstrated that BR is tangent to the circle at point B, using the fact that PR is parallel to AQ and the inherent right angles formed by the tangents and radii of the circle.