Question 1177001: EVOL is a cyclic quadrilateral, inscribed in a circle with center S. Given that the radius of this circle is 25 in. and angle VEL is 55 degrees, explain your work to find the following measurements.
a)Length of Major Arc VL
b)Angle measure of Minor Arc VL
c)Angle VOL
d)Length of Chord VL
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely, let's break down this geometry problem step-by-step.
**Understanding the Setup**
* **EVOL is a cyclic quadrilateral:** This means all four vertices (E, V, O, L) lie on the circumference of a circle.
* **Center S:** The center of the circle is denoted as S.
* **Radius = 25 in:** The distance from S to any point on the circle is 25 inches.
* **∠VEL = 55 degrees:** This is an inscribed angle.
**Solving the Problem**
**a) Length of Major Arc VL**
1. **Angle VOL (Central Angle):**
* In a cyclic quadrilateral, opposite angles are supplementary (add up to 180 degrees). However we do not need that fact for this part of the problem.
* The measure of a central angle is twice the measure of an inscribed angle that subtends the same arc.
* Therefore, ∠VOL = 2 * ∠VEL = 2 * 55 degrees = 110 degrees. This is the measure of the minor arc VL.
* The major arc VL is 360 degrees - 110 degrees = 250 degrees.
2. **Circumference:**
* The circumference of the circle is C = 2 * π * r = 2 * π * 25 = 50π inches.
3. **Length of Major Arc VL:**
* The length of the major arc is the fraction of the circumference corresponding to the central angle of the major arc.
* Length of major arc VL = (250/360) * 50π = (25/36) * 50π = (1250π/36) = (625π/18) inches.
**b) Angle Measure of Minor Arc VL**
* As calculated in part (a), the measure of minor arc VL is equal to the central angle that subtends it, which is ∠VOL.
* Therefore, the angle measure of minor arc VL is 110 degrees.
**c) Angle VOL**
* We already calculated this in part (a).
* ∠VOL = 110 degrees.
**d) Length of Chord VL**
1. **Triangle VOL:**
* Triangle VOL is an isosceles triangle because SV = SL = 25 inches (both are radii).
* We know angle VOL is 110 degrees.
2. **Law of Cosines:**
* We can use the Law of Cosines to find the length of chord VL:
* VL² = VO² + LO² - 2 * VO * LO * cos(∠VOL)
* VL² = 25² + 25² - 2 * 25 * 25 * cos(110°)
* VL² = 625 + 625 - 1250 * cos(110°)
* VL² = 1250 - 1250 * (-0.3420)
* VL² = 1250 + 427.5
* VL² = 1677.5
* VL = √1677.5 ≈ 40.96 inches.
**Summary of Answers**
* **a) Length of Major Arc VL:** 625π/18 inches (approximately 109.08 inches)
* **b) Angle Measure of Minor Arc VL:** 110 degrees
* **c) Angle VOL:** 110 degrees
* **d) Length of Chord VL:** Approximately 40.96 inches
Answer by ikleyn(52803)  (Show Source):
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