SOLUTION: 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 follo

Question 1176989: 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!
You got it! Let's go through the problem again, with a focus on clear explanations.
**Understanding the Setup**
* **EVOL is a cyclic quadrilateral:** This means that all four points (E, V, O, L) lie on the circumference of a circle.
* **Center S:** The point S is the exact center of the circle.
* **Radius = 25 inches:** The distance from the center S to any point on the circle's edge (E, V, O, or L) is 25 inches.
* **Angle VEL = 55 degrees:** This is an inscribed angle, meaning its vertex (E) lies on the circle.
**Solving the Problem**
**a) Length of Major Arc VL**
1. **Find Angle VOL (The Central Angle):**
* A central angle is an angle formed by two radii of a circle, with its vertex at the center.
* The measure of a central angle is twice the measure of an inscribed angle that intercepts the same arc.
* Since angle VEL intercepts arc VL, and angle VOL also intercepts arc VL, we have:
* Angle VOL = 2 * Angle VEL = 2 * 55 degrees = 110 degrees.
* This 110 degrees is the measure of the *minor* arc VL.
* The *major* arc VL is the rest of the circle. So:
* Major arc VL = 360 degrees - 110 degrees = 250 degrees.
2. **Calculate the Circumference:**
* The circumference (the total distance around the circle) is found using the formula:
* Circumference = 2 * π * radius
* Circumference = 2 * π * 25 inches = 50π inches.
3. **Find the Length of the Major Arc:**
* The length of an arc is a fraction of the circle's circumference, determined by the arc's central angle.
* Length of Major Arc VL = (Central Angle of Major Arc / 360 degrees) * Circumference
* Length of Major Arc VL = (250 degrees / 360 degrees) * 50π inches
* Length of Major Arc VL = (25/36) * 50π inches = 1250π/36 inches = 625π/18 inches.
* This is approximately 109.08 inches.
**b) Angle Measure of Minor Arc VL**
* The angle measure of a minor arc is the same as the measure of the central angle that intercepts it.
* Therefore, the angle measure of minor arc VL is 110 degrees.
**c) Angle VOL**
* We already calculated this in part (a).
* Angle VOL = 110 degrees.
**d) Length of Chord VL**
1. **Visualize Triangle VOL:**
* Connect points V and L with a straight line (the chord VL).
* Triangle VOL is formed by the radii SV and SL, and the chord VL.
* Because SV and SL are radii, they are equal in measure, therefore triangle VOL is an isosceles triangle.
2. **Use the Law of Cosines:**
* The Law of Cosines is used to find a side length in a triangle when you know two side lengths and the included angle.
* In triangle VOL:
* VO = 25 inches (radius)
* LO = 25 inches (radius)
* Angle VOL = 110 degrees
* The Law of Cosines formula is: VL² = VO² + LO² - 2 * VO * LO * cos(Angle VOL)
* VL² = 25² + 25² - 2 * 25 * 25 * cos(110 degrees)
* VL² = 625 + 625 - 1250 * (-0.3420) (approximately)
* VL² = 1250 + 427.5 = 1677.5
* VL = √1677.5 ≈ 40.96 inches.
**Summary**
* **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(52782) (Show Source): You can put this solution on YOUR website!
.

The answer to question (c) in the post by @CPhill, giving angle VOL = 110°, is INCORRECT.

The correct answer to this question is 125°, which complements the angle VEL of 55° to 180°.

It is the same correct answer as Edwin produced to this problem in his post.
https://www.algebra.com/algebra/homework/Length-and-distance/Length-and-distance.faq.question.1177016.html

For any quadrilateral, inscribed in a circle, its opposite angles are supplementary:
they complement each other to 180°.

This property of quadrilaterals inscribed into a circle is a standard geometric statement
which students learn/study in a standard Geometry curriculum.

See, for example, my lesson

A property of the angles of a quadrilateral inscribed in a circle

https://www.algebra.com/algebra/homework/word/geometry/The-property-of-the-angles-of-a-quadrilateral-inscribed-in-a-circle.lesson>

in this site.

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