Answer:
Area of trapezium = 4.4132 R²
Step-by-step explanation:
Given, MNPK is a trapezoid
MN = PK and ∠NMK = 65°
OT = R.
⇒ ∠PKM = 65° and also ∠MNP = ∠KPN = x (say).
Now, sum of interior angles in a quadrilateral of 4 sides = 360°.
⇒ x + x + 65° + 65° = 360°
⇒ x = 115°.
Here, NS is a tangent to the circle and ∠NSO = 90°
consider triangle NOS;
line joining O and N bisects the angle ∠MNP
⇒ ∠ONS = [tex]\frac{115}{2}[/tex] = 57.5°
Now, tan(57.5°) = [tex]\frac{OS}{SN}[/tex]
⇒ 1.5697 = [tex]\frac{R}{SN}[/tex]
⇒ SN = 0.637 R
⇒ NP = 2×SN = 2× 0.637 R = 1.274 R
Now, draw a line parallel to ST from N to line MK
let the intersection point be Q.
⇒ NQ = 2R
Consider triangle NQM,
tan(∠NMQ) = [tex]\frac{NQ}{QM}[/tex]
⇒ tan65° = [tex]\frac{NQ}{QM}[/tex]
⇒ QM = [tex]\frac{2R}{2.1445}[/tex]
QM = 0.9326 R .
⇒ MT = MQ + QT
= 0.9326 R + 0.637 R (as QT = SN)
⇒ MT = 1.5696 R
⇒ MK = 2×MT = 2×1.5696 R = 3.1392 R
Now, area of trapezium is (sum of parallel sides/ 2)×(distance between them).
⇒ A = ([tex]\frac{NP + MK}{2}[/tex]) × (ST)
= ([tex]\frac{1.274 R + 3.1392 R}{2}[/tex]) × 2 R
= 4.4132 R²
⇒ Area of trapezium = 4.4132 R²
