Mada za sehemu hiiCirclesMada 5
A Tangent to a Circle
A tangent is a line which touches a circle at exactly one point. The point where the line touches the circle is called the point of contact. A tangent is always perpendicular to the radius drawn to the point of contact.
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- A common tangent is a line that is a tangent to each of two circles.
- A common external tangent does not intersect the line segment joining the centers of the circles.
- A common internal tangent intersects the line segment joining the centers of the circles.
Theorem 1: Intersecting Chords Theorem
If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the segments of the other chord.
Intersecting Chords Rule: (segment piece) × (segment piece) = (segment piece) × (segment piece)
Theorem Proof:
Proof Using Two-Column Table
| Statements | Reasons |
|---|---|
| 1. Chords AB and CD | 1. Given |
| 2. Draw lines AC and BD | 2. Two points determine only one line. |
| 3. ∠A ≅ ∠C and ∠B ≅ ∠D | 3. If two inscribed angles intercept the same arc, the angles are congruent. |
| 4. △ADE ∼ △CBE | 4. AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. |
| 5. AE / CE = EB / BD | 5. Corresponding sides of similar triangles are in proportion. |
| 6. AB × EB = CE × ED | 6. In a proportion, the product of the means equals the product of the extremes. |
Theorem 2: Secant-Secant Theorem
If two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.
Secant-Secant Rule: (whole secant) × (external part) = (whole secant) × (external part)
Theorem 3: Secant-Tangent Theorem
If a secant segment and a tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.
Secant-Tangent Rule: (whole secant) × (external part) = (tangent)²
Example 1
Two common tangents to a circle form a minor arc with a central angle of 140°. Find the angle formed between the tangents.
Solution
Two tangents drawn from the same external point to a circle are equal in length and form two right angles (90° each) with the radii. Together with the central angle, these form a quadrilateral with a total angle sum of 360°.
Let y be the angle formed between the two tangents.
Then:
y + 90° + 90° + 140° = 360°
y + 180° + 140° = 360°
y + 320° = 360°
y = 40°
Therefore, the angle formed between the tangents is 40 degrees.
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