Question

Line PQ is tangent to the circle at point A. Arc AB congruent AC. Complete the following activity to prove triangle ABC as an isosceles triangle ​

Answer 1

Step-by-step explanation:

In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then

(1) What is the length of each tangent segment ?

(2) What is the measure of ∠MRO ?

(3) What is the measure of ∠ MRN ?

ANSWER:

(1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively.

∴ ∠OMR = ∠ONR = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

OM = 5 cm and OR = 10 cm

In right ∆OMR,

OR2=OM2+MR2⇒MR=OR2−OM2−−−−−−−−−−√ ⇒MR=102−52−−−−−−−√⇒MR=100−25−−−−−−−√=75−−√=53–√ cm

Tangent segments drawn from an external point to a circle are congruent.

∴ MR = NR = 53–√ cm

(2) In right ∆OMR,

tan∠MRO=OMMR⇒tan∠MRO=5 cm53√ cm=13√⇒tan∠MRO=tan30°⇒∠MRO=30°

Thus, the measure of ∠MRO is 30º.

Similarly, ∠NRO = 30º

(3) ∠MRN = ∠MRO + ∠NRO = 30º + 30º = 60º

Thus, the measure of ∠MRN is 60º.