Question

In the diagram of circle O, m NQM  103

. If the measure of LP is 30

more than the measure of MN,

then determine the measure of MN. Show how you arrived at your answer.

Answer 1

88 degree

Explanation:

We assume the measure of MN is x degree.

As the measure of LP is 30 degree more than that of MN, so that the measure of LP is: x + 30 degree

In the circle, as 4 points M,N,P,L are on the circle, we have:

+) ∡MPN = 1/2 * (measure of ∡MPQMN) = x/2 = ∡MPQ

+) ∡LMP =1/2 * (measure of LP) = (x+30)/2 = ∡QMP

We have ∡NQM and ∡MQP are complementary angles, so that:

∡MQP + ∡NQM = 180

=> ∡MQP = 180 - ∡NQM = 180 -103 = 77

In the triangle QMP, total measure of 3 internal angles are 180 degree, so that:

∡MQP + ∡QMP + ∡MPQ = 180

=> 77 + (x + 30)/2 + x/2 = 180

=> 77 + x/2 + 15 + x/2 = 180

=> x = 180 -77-15= 88

So that the measure of MN is 88 degree