How Do You Prove That A Tangent To A Circle Is Perpendicular: Geometry Demystified

## Proof: Perpendicular To Radius Is Tangent To Circle | Geometry

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## How Do You Prove That Tangent Form 90 Degrees?

To understand how to demonstrate that a tangent forms a 90-degree angle with a circle, it’s essential to grasp the concept of a tangent line. A tangent line is a line that touches a circle at a single point on its outer boundary. If we extend this idea by drawing a radius that originates from the circle’s center and intersects the circle at the same point where the tangent touches, an important relationship becomes apparent. The key insight here is that the angle formed between this radius and the tangent line will always measure exactly 90 degrees. This fundamental geometric principle helps establish the connection between tangents and right angles when they intersect a circle. Please note that this explanation is accurate as of September 2021.

## How Do You Prove A Tangent To A Circle?

How can we establish the validity of a tangent line to a circle? To demonstrate this geometric concept, we will employ the Circle Theorem Proof for the Length of Tangents, as illustrated in a YouTube video. In this proof, let’s begin by considering points A and B, which correspond to the endpoints of the tangent line touching the circle. Given that both OA and OB are radii of the circle, we can confidently assert that OA is equal in length to OB, denoting this shared length as X. Now, if we connect point C to the center of the circle, O, we form two distinct triangles. In this context, the length of OC can be designated as Y. This comprehensive explanation will help you grasp the fundamental principles behind proving the existence of tangents to a circle. For a visual demonstration and further insights, you can watch the YouTube video linked above.

## Summary 34 How do you prove that a tangent to a circle is perpendicular

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⇒ OA < OB (since OA = OC). The same will be the case for all other points on the tangent (l). So **OA is shorter than any other line segment joining O to any point on l**. Hence, the tangent at any point of a circle is perpendicular to the radius through the point of contact.A tangent is a line that just touches the circle at a single point on its circumference. **If we draw a radius that meets the circumference at the same point, the angle between the radius and the tangent will always be exactly 90°**.**At point P, a tangent PR has been drawn touching the circle.** **From point Q lies on the circle, draw QP ⊥ RP at point P.** **Since ∠OPR = ∠QPR (each 90°), it is possible only when centre O lies on the line QP**. Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Learn more about the topic How do you prove that a tangent to a circle is perpendicular.

- Theorem – The tangent at any point of a circle is perpendicular …
- Tangent and radius of a circle meet at 90 – GraphicMaths
- Circle Theorem Proof – Length of Tangents Proof – YouTube
- Prove that the perpendicular at the point of contact to the tangent to a …
- Tangent lines to circles – Wikipedia
- Radii to Tangents – Concept – Geometry Video by Brightstorm

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