Questions: จากรูปกำหนดให้ วงกลม C มี X Ĉ Z=50°, Y X̂ Z=25° และ XC=CZ เป็นรัศมี จงหา 1. ขนาดของ x Ô Z 2. ขนาดของ X Ŷ Z

จากรูปกำหนดให้ วงกลม C มี X Ĉ Z=50°, Y X̂ Z=25° และ XC=CZ เป็นรัศมี จงหา 1. ขนาดของ x Ô Z 2. ขนาดของ X Ŷ Z
Transcript text: จากรูปกำหนดให้ วงกลม $C$ มี $X \hat{C} Z=50^{\circ}, Y \widehat{X Z}=25^{\circ}$ และ $\overline{X C}=\overline{C Z}$ เป็นรัศมี จงหา 1. ขนาดของ $x \widehat{O} Z$ 2. ขนาดของ $X \hat{Y} Z$
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Solution

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Solution Steps

Step 1: Find the measure of angle XOZ

Since XC = CZ are radii, triangle XCZ is isosceles. Therefore, angle CXO = angle CZO. Also, the sum of angles in triangle XCZ is 180°. So, angle CXO + angle CZO + angle XCZ = 180°. We know angle XCZ = 50°. Therefore, 2 * angle CXO + 50° = 180°. This gives 2 * angle CXO = 130°, and angle CXO = 65°. Angle XOZ is twice the inscribed angle CXO, so angle XOZ = 2 * 65° = 130°.

Step 2: Find the measure of angle XYZ

Angle XCZ is the central angle subtended by arc XZ. Angle XYZ is an inscribed angle subtended by the same arc XZ. The measure of an inscribed angle is half the measure of the central angle subtended by the same arc. Therefore, angle XYZ = angle XCZ / 2 = 50°/2 = 25°.

Final Answer

  1. 130°
  2. 25°
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