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7.7 Segments of chords secants and tangents

Theorem : If two chords, seg.AB and seg.CD intersect inside or outside a circle at P then l (seg. PA) ´ l (seg. PB) = l (seg. PC) ´ l (seg. PD)

Figure 7.21 (a)             Figure 7.21 (b)

In figure 7.21 (a) P is in the interior of the circle. Join AC and BD and consider DAPC and DBDP.

m Ð APC = m Ð BPD (vertical angles).

m Ð CAP = m Ð BDP (angles inscribed in the same arc).

\ DAPC ~ D BPD ( A A test )

\ ( corresponding sides of similar triangles).

l (seg. PA) ´ l (seg. PB) = l (seg. PC) ´ l (seg. PD).

Now consider figure 7.21 (b).

P is in the exterior of the circle. Join A to C and B to D.

Consider D PAC and D PBD

m Ð APC = m Ð BPD (same angle).

m Ð CAP = m Ð PDB (exterior angle property of a cyclic quadrilateral).

\ D PAC ~ D PDB ( A A test )

\ (corresponding sides of similar triangles).

\ l (seg. PA) ´ l (seg. PB) = l (seg. PC) ´ l (seg. PD).

Consider a secant PAB to a circle, (figure 7.22) intersecting the circle at A and B and line PT is a tangent then l (seg. PA) ´ l (seg. PB) = l (seg. PT)2.

Figure 7.22

P is a point in the exterior of the circle. A secant passes through P and intersects the circle at points A & B. Tangent through P touches the circle in point T.

To prove that l (seg. PA) ´ l (seg. PB) = l (seg. PT)2

Consider D PTA and D PTB.

m Ð TPA = Ð TPB ( same angle)

According to Tangent Secant theorem,

m Ð ATP = m (arc AT)

= m Ð PTB ( inscribed angle )

\ D PTA ~ D PTB ( A A test )

\ (corresponding sides of similar triangles).

\ l (seg. PA) ´ l (seg. PB) = l (seg. PT)2.

Theorem: The lengths of two tangent segments from an external point to a circle are equal.

As shown in figure 7.23 seg. QR and seg. QS are two tangents on a circle with P as its center.

Figure 7.23

To prove that l (seg.QR) = l (seg.QS) join P to Q and R to S.

m Ð PRQ = m Ð PSQ = 900.

The radius and the tangent form a right angle at the point of tangency,

\ D PRQ and D PSQ are right triangles such that

seg. PR @ seg PS (radii of the same circle).

seg. PQ @ seg. PQ (same side).

\ D PRQ @ D PSQ (H.S)

\ seg.QR @ seg.QS (corresponding sides of congruent triangles are congruent).

\ l (seg.QR) = l (seg.QS).

Index

7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed angels
7.5 Some properties od tangents, secants and chords
7.6 Chords and their arcs
7.7 Segments of chords secants and tangents
7.8 Lengths of arcs and area of sectors

Chapter 8

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