Q1 of 26 Page 172

In the picture, the sides of the large triangle are tangents to the circumcircle of the small triangle, through its vertices.


Calculate the angles of the large triangle.

Let us label the diagram.



We know, By alternate segment theorem the line joining the centre and the points where the tangents meet bisects the angle between the radii.


⇒ ∠ ABQ = ∠ACB and ∠BAQ = ∠ACB [AB is a chord]


⇒ ∠ABQ = ∠BAQ = ∠60° [∠ACB = 60°]


In ΔABQ, By triangle sum property


⇒ ∠ABQ + ∠BAQ + ∠AQB = 180°


⇒ 60° + 60° + ∠Q = 180°


⇒ ∠Q = 60°


Also, By alternate segment theorem


∠ACP = ∠ ABC and ∠CAP = ∠ABC [AC is a chord]


⇒ ∠ACP = ∠CAP = 80° [∠ABC = 80°]


In ΔACP, By triangle sum property


⇒ ∠ACP + ∠CAP + ∠APC = 180°


⇒ 80° + 80° + ∠P = 180°


⇒ ∠P = 20°


In ΔPQR, By angle sum property


∠P + ∠Q + ∠R = 180°


⇒ 20° + 60° + ∠R = 180°


⇒ ∠R = 100°


Hence, three angles are 20°, 60° and 100°.


More from this chapter

All 26 →
3

The picture shows the tangents at two points on a circle and the radii through the points of contact.

Prove that this line is the perpendicular bisector of the chords joining the points of contact.


 4

Prove that the quadrilateral with sides as the tangents at the ends of a pair of perpendicular chords of a circle is cyclic.

What sort of a quadrilateral do we get if one chord is a diameter? And if both chords are diameters?


 2

In the picture, the sides of the large triangle are tangents of the circum-circle of the smaller triangle, through its vertices.


Calculate the angles of the smaller triangle.

 3

In the picture, PQ, RS, TU are tangents to the circumcircle of Δ ABC.

Sort out the equal angles in the picture.