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Some properties of tangents, secants and chords, Angles formed by intersecting chords, tangent and chord and two secants, Chords and their arcs, Segments of chords secants and tangents, Lengths of arcs and areas of sectors
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Some properties of tangents, secants and chords
A line in the plane of the circle that intersects the circle at exactly one point is called tangent line. The point of intersection is called the point of tangency.
The Tangent- Line TheoremIf a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint. OrIf the tangent to a circle and the radius of the circle intersect they do so at right angles :
Some properties of tangents, secants and chords
The Tangent- Line Theorem
If a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint. Or
If the tangent to a circle and the radius of the circle intersect they do so at right angles :
Some properties of tangents, secants and chords
Example 1.Line AP is tangent to circle C at P. If seg. CP=7cm and seg. AP=24cm, how far is point A from a center C?
Some properties of tangents, secants and chords
Corollary:
The tangents to a circle at the endpoints of a diameter are parallel.
If line l and line m are perpendicular to a diameter EP, then line l is parallel to line m.
Some properties of tangents, secants and chords
The Tangent-Segments Theorem
1. The lengths of two tangent segments from an external point to a circle are equal.
2. The angles between the tangent segments and the line joining the external point to the center of the circle are congruent.
1. If line EF and line PF are tangent to circle C at E and P respectively intersects at F, then EF = PF.
Some properties of tangents, secants and chords
The Tangent-Segments Theorem
1. The lengths of two tangent segments from an external point to a circle are equal.
2. The angles between the tangent segments and the line joining the external point to the center of the circle are congruent.
2. If tangent segments EF and PF form an angle EFP at F, then angle EFC = angle CFP.
Angles formed by intersecting chords
If two chords intersect in a circle, the angle they form is half the sum of the intercepted arcs.
Angles formed by intersecting chords
Example 2.
Find the value of x.
Angles formed by intersecting tangent and chord
m∠𝐻𝑃𝐾=12𝑎𝑟𝑐 𝐻𝑃
Tangent Secant Theorem
If a chord intersects the tangent at the point of tangency, the angle it forms is half the measure of the intercepted arc.
Angles formed by intersecting tangent and chord
Tangent Secant Theorem
Example 3.Find the value of x.
Angles formed by intersecting two secants
If two secants intersect outside a circle half the difference in the measures of the intercepted arcs gives the angle formed by the two secants.
𝑚∠𝑞=12
(𝑎𝑟𝑐 𝑃𝑄−𝑎𝑟𝑐𝑂𝑅 )
Angles formed by intersecting two secants
Example 4 Find the value of x.
Example 5
Match the figure with the formula below.
b¿
a ¿
d ¿
c ¿
2.
3.
4.
Chords and their arcs
Theorem: If in any circle two chords are equal in length then the measures of their corresponding minor arcs are same.
Chords and their arcs
Theorem: if two chords are equidistant from the center of the circle, they are equal in measure.
and
Chords and their arcs
Theorem: The perpendicular from the center of a circle to a chord of the circle bisects the chord.
In the figure, XY is the chord of a circle with center O. Seg.OP is the perpendicular from the center to the chord. According to the theorem given above seg XP = seg. YP.
Example 6
a. Find the degree measure of each of the five congruent arcs of a circle around a regular pentagon.
b. Solve the length of x.
Segments of chords, secants and tangents
The Intersecting segments of Chords TheoremIf two chords are intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
𝑀𝑃 ∙𝑃𝑂=𝑁𝑃 ∙𝑃𝐿
Segments of chords, secants and tangents
Example 7
Solve the value of x.
Segments of chords, secants and tangents
The Segments of Secants Theorem
If two secants intersect in the exterior of the circle, the product of the length of one secant segment and the length of its external part is equal to the product of the length of the other secant and the length of its external part. QU ∙𝑄𝑅=𝑄𝑇 ∙𝑄𝑆
Segments of chords, secants and tangents
Example 8
Solve the value of x.
Segments of chords, secants and tangents
The Tangent Secant Segments Theorem
If a tangent segment and a secant segment intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part. 𝑃𝑇 2=𝐵𝑃 ∙ 𝐴𝑃
Example 9
PT is a tangent intersecting the secant through AB at P. Given (seg. PA) = 2.5 cm. and (seg.AB)=4.5 cm., find (seg PT).
Segments of chords, secants and tangents
Perimeter or circumference of a circle = where r is the radius or C=, where d is the diameter d=2r
Perimeter and Area of a Circle
Area of a circle = r2 where ' r ' is the radius of the circle
Area of a circle = 2 where ' d ' is the diameter of the circle
Example 10
a. Find the circumference of a circle with area 25 sq. ft.
Perimeter and Area of a Circle
b. Find the area of a circle with circumference 30 cm.
An arc is a part of the circumference of the circle; a part proportional to the central angle.If 3600 corresponds to the full circumference. i.e. 2r then for a central angle of m0 the corresponding arc length will be l such that
Lengths of arcs and areas of sectors
Analogically consider the area of a sector. This too is proportional to the central angle. 3600 corresponds to area of the circle r2. Therefore for a central angle the area of the sector will be in the ratio
Lengths of arcs and areas of sectors
Example 11
In a circle with the radius of 2 cm, the central angle for an arc AB is 750. Find (seg.AB). Also find the area of the sector AOB having a central angle of 750
Lengths of arcs and areas of sectors
