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Solve for Unknown Angle x in a Circle with Diameter as Triangle Side
Mathematics
Grade 9 (Junior High School)
Question Content
In circle C, the triangle inscribed in the circle has one angle of 68°, and the side opposite the unknown angle x° is the diameter of the circle. What is the value of x?
Correct Answer
22°
Detailed Solution Steps
1
Step 1: Recognize that the side opposite the unmarked angle (pointed by the red arrow) is the diameter of circle C. By the Thales' theorem, an angle inscribed in a semicircle is a right angle, so this unmarked angle is 90°.
2
Step 2: Recall that the sum of the interior angles of a triangle is 180°. Set up the equation: x° + 68° + 90° = 180°.
3
Step 3: Calculate the sum of the known angles: 68° + 90° = 158°.
4
Step 4: Solve for x: x° = 180° - 158° = 22°.
Knowledge Points Involved
1
Thales' Theorem (Angle in a Semicircle Theorem)
This theorem states that if an angle is inscribed in a semicircle (i.e., the side opposite the angle is the diameter of the circle), then the angle is a right angle (90°). It is a special case of the inscribed angle theorem, used to identify right angles in circle geometry problems.
2
Sum of Interior Angles of a Triangle
The sum of the three interior angles of any triangle is always 180°. This fundamental property is used to find unknown angles in a triangle when two other angles are known.
3
Inscribed Angles in Circles
An inscribed angle is an angle formed by two chords in a circle with the vertex on the circle. Inscribed angles subtended by the same arc are equal, and an inscribed angle is half the measure of its subtended central angle. Thales' theorem is a specific application of this concept.
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