Math Problem Statement

Consider the circumscribed angle. Determine the value of x.

Solution

To solve this problem, we need to analyze the geometry of the circumscribed angle.

The problem shows a circumscribed angle formed by the tangents to a circle, with the angle's measure given as 2929^\circ, and one of the intercepted arcs labeled as 2x+52x + 5.

Key Concept:

In a circle, the measure of a circumscribed angle (an angle formed outside the circle by two tangents) is half the difference between the measures of the intercepted arcs.

Given:

  • The circumscribed angle is 2929^\circ.
  • One of the arcs intercepted by the angle is labeled 2x+52x + 5 (degrees).

Let's denote:

  • The measure of the other arc as yy (which is the arc opposite to 2x+52x + 5).

Since the angle is formed by tangents, the total measure of both arcs should add up to 360360^\circ because the entire circle's arc measures sum to 360360^\circ.

Thus, we have: y+(2x+5)=360y + (2x + 5) = 360 So, y=360(2x+5)=3552xy = 360 - (2x + 5) = 355 - 2x

Applying the Circumscribed Angle Formula:

The measure of the circumscribed angle is: Circumscribed angle=12((2x+5)y)\text{Circumscribed angle} = \frac{1}{2} \left( (2x + 5) - y \right)

Substitute y=3552xy = 355 - 2x: 29=12((2x+5)(3552x))29 = \frac{1}{2} \left( (2x + 5) - (355 - 2x) \right) 29=12(2x+5355+2x)29 = \frac{1}{2} \left( 2x + 5 - 355 + 2x \right) 29=12(4x350)29 = \frac{1}{2} (4x - 350) Multiply both sides by 2: 58=4x35058 = 4x - 350 Add 350 to both sides: 408=4x408 = 4x Divide by 4: x=102x = 102

However, this does not match any of the answer choices, indicating there might be a mistake in setup.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circumscribed Angle
Circle Properties

Formulas

Circumscribed angle = (1/2) * (larger intercepted arc - smaller intercepted arc)

Theorems

Circumscribed Angle Theorem

Suitable Grade Level

Grades 9-10