急! Properties of Circles 12q7

10) With △PQR and △PRT being a pair of congurent isos. triangles (SAS):

Let ∠QPR = TPR = x, then

∠PQR = ∠PRQ = ∠PRT = ∠PTR = (180 - x)/2

Hence ∠QRT = ∠PRQ + ∠PRT = 180 - x

Thus ∠QRS = x = ∠QPR

Hence ST is a tangent to the circle at R (Converse of angle in the same segment)

13a) Join AC, then ∠BAC = 72 (∠ in alt. segment)

Let ∠TAB = x, then ∠BAC = x (∠ in alt. segment)

In △TAC, x + 72 + x + 38 = 180

2x = 70

x = 35

b) ∠ABC = 180 - ∠ACB - ∠BAC = 180 - 35 - 72 = 73

7) ∠DOB = 180 - 42 = 138 (Opp. ∠s, cyclic quad.)

∠BAD = 1/2 x ∠DOB = 69 (∠ at centre = Twice ∠ at circum)

8a) EF touches both circles at T

Join PT and QT, both of them are perp. to EF.

Thus P, T, Q is a straight line.

b) Radius of smaller circle = 6 cm

Hence radius of larger circle = 10 cm

Thus PQ = 16 cm