Page 163 - Mathematics_Form_3

Circles

Proof: Hence, ∠XZY = 50° .
ˆ
FDB 90= ° (angle in the semicircle).
Similarly,
DFB DBF 90+ ˆ ˆ = ° (sum of interior
angle of FBD∆ ). ∠ZXY = 75° and ∠XYZ = 55°.
FOR ONLINE READING ONLY
DBF DBC 90+ ˆ ˆ = ° (the radius is Therefore, the angles of ∆XZY are
50°, 75°, and 55° (angles formed in
perpendicular to the tangent)
the alternate segments of a circle are
Thus, equal).
DFB DBF DBF DBC+ DFB DBF DBF+ ˆ ˆ ˆ ˆ = ˆ ˆ + + = ˆ ˆ →DBC → DFB DBC= DFB DBC= ˆ ˆ ˆ ˆ

DFB DBF DBF DBC+ ˆ ˆ = ˆ + ˆ ⇒ DFB DBC→ ˆ = ˆ Intersecting secant and tangent
ˆ
ˆ
BED = DFB (angle in the same
You have already established the
segment) relationship between line segments
ˆ
ˆ
Therefore, DBC = BED. when chords intersect inside and outside
the circle.
Example 5.24
This section introduces the relationship
In the following figure, a circle is between line segments when a secant
inscribed in ∆ABC touching it at X, and a tangent meet at a point.
Y, and Z. If the angles of ∆ABC are
Engage in Activity 5.17 to learn
70°, 80°, and 30°, find the angles of
the relationship between lengths of
triangle XYZ.
intersecting secant and tangent.
B
70º Activity 5.17: Deducing relationship
Z
between lengths of intersecting
Mathematics for Secondary Schools Solution Y 30º C 2. Draw a tangent AT to a circle and
secant and tangent
X
1. Draw a circle of any convenient
80º
radius with centre at O.
A

point A as a tangency point.
From the figure, it implies that
ΔAXY is isosceles (tangents from a

3. Draw a secant BT to intersect the
common external point are equal).
tangent at T. Label the point where
Thus,
the secant intersect the circle as C.
1 _
∠AXY = (180° − 80° ) = 50° 4. Using a ruler or otherwise,
2
∠XZY = ∠AXY (alternate segment measure the lengths of AT,BT ,

theorem). and CT

156 Student\s Book Form Three

18/09/2025 09:59:54
MATHEMATIC F3 SB.indd 156
MATHEMATIC F3 SB.indd 156 18/09/2025 09:59:54