Answer:
arc HD = 20°
Step-by-step explanation:
Let point N be the point where secant RL intersect the circle, point M be the point of intersection for chords NL and KH, and point S be the vertex of the triangle formed by secants KH and RD.
It is given that m∠SRM = 38∘ and m∠RMS = 92∘. Use the Triangle Sum Theorem to determine m∠MSR.
m∠MSR = 180° − 92∘ − 38∘ = 50∘
It is also given that mPN = 18∘ and mNK = 102∘. So, mPK = 18∘ + 102∘ = 120∘.
If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. So,
m∠MSR= 1 /2 (mPK − mHD)
Substitute the known values and solve for mHD.
50∘ = 1/2 (120∘ − mHD)
Multiply by 2.
100 = 120∘ − mHD
Simplify.
mHD = 20
Therefore, the measure of arc HD is 20∘.
