1. Statement of problem: Find the radius, diameter, center, and other circle parts as labeled, then solve algebraic problems involving central and inscribed angles. 2. Definitions: - Radius: segment from center O to a point on the circle (e.g., OB). - Diameter: chord passing through the center (e.g., AC). - Center: point O. - Interior point: point inside circle but not center (e.g., E). - Exterior point: point outside circle (e.g., F). - Chord: segment joining two points on the circle (e.g., AB). - Secant line: line crossing circle in two points (line through D). - Tangent line: line touching circle at one point (at C). - Point of tangency: point C. 3. Algebra problems: 21. Given: m∠AOC=85°, AC=5x. Since ∠AOC is a central angle, its intercepted arc AC has measure equal to 85°. So, mAC = 85 = 5x Solve for x: $$5x=85$$ $$x=\frac{85}{5}=17$$ Answer: x=17 22. Chords AE and BF are congruent. Distance from center to AE = x+4 Distance from center to BF = 3x-4 Since chords are congruent, their distances from center are equal: $$x+4 = 3x -4$$ $$4+4=3x - x$$ $$8=2x$$ $$x=4$$ Find chord lengths: We don't have chord length formula here, but since distances are equal, the chord lengths are equal. Without length formula, length cannot be numerically found. Answer: The chords AE and BF have equal length but value cannot be determined without more info. 23. ∠DOC is inscribed angle intercepting DC. mDC = 148°, m∠DOC = 5x -2 The measure of an inscribed angle is half the measure of its intercepted arc: $$m\angle DOC = \frac{1}{2} mDC$$ $$5x -2 = \frac{148}{2} = 74$$ $$5x = 74 + 2 = 76$$ $$x = \frac{76}{5} = 15.2$$ Find m∠DOC: $$5(15.2) -2 = 76 - 2 = 74°$$ Answer: m∠DOC = 74° 24. ∠ABC and ∠AFC intercept the same arc. Given m∠ABC = 7x -5, m∠AFC = 3x + 19. Since intercept same arc, angles are equal: $$7x -5 = 3x +19$$ $$7x -3x = 19 +5$$ $$4x = 24$$ $$x=6$$ Find angles: $$7(6) -5 = 42 -5 = 37°$$ $$3(6) +19 = 18 + 19 = 37°$$ Answer: Each angle measures 37° 25. ∠XYZ intercepts semicircle. For angle that intercepts a semicircle, the angle is a straight angle (180°). Given m∠XYZ = 2x + 6 So: $$2x + 6 = 180$$ $$2x = 174$$ $$x = 87$$ Answer: x = 87 26. Opposite angles ∠M and ∠O in quadrilateral inscribed in circle are supplementary. Given m∠M = 3x +4, m∠O = 2x + 6 Since supplementary: $$m∠M + m∠O = 180$$ $$(3x +4) + (2x +6) = 180$$ $$5x +10=180$$ $$5x=170$$ $$x=34$$ Find angles: $$m∠M=3(34)+4=102 +4 =106°$$ $$m∠O=2(34)+6=68 +6 =74°$$ Answer: m∠M=106°, m∠O=74° 27. Chords ST and UV intersect at point X inside circle. Given SX=5, XT=8, UV=10, XV=3x By chord segment theorem: $$SX imes XT = VX imes XU$$ But VX = 3x, UV=10 then XU = UV - XV = 10 - 3x So: $$5 imes 8 = 3x (10 - 3x)$$ $$40 = 30x - 9x^{2}$$ Rearranged: $$9x^{2} - 30x + 40 = 0$$ Divide entire eq by 1 (no simplification): solve quadratic: Discriminant: $$\Delta = (-30)^{2} - 4 \times 9 \times 40 = 900 - 1440 = -540 <0$$ No real solution=> check data or problem setup. Assuming typo, maybe XV=x not 3x; if XV=x: $$5 imes 8 = x (10 - x)$$ $$40 = 10x - x^{2}$$ $$x^{2} -10x +40=0$$ Discriminant: $$\Delta=100 -160 = -60 <0$$ No real roots either. Possibly problem expects algebraic form answer only. Answer: No real solution with given data. 28. Given m∠SXV=85°, m∠SV=100°, m∠TU=4x. No direct relation given: must relate angle sum or supplementary. Assuming angles in triangle or quadrilateral add 180 or 360. Check if m∠SXV + m∠SV + m∠TU = 180: $$85 + 100 + 4x =180$$ $$185 +4x=180$$ $$4x= -5$$ $$x= -1.25$$ Negative x unlikely. Possibility: missing problem details. Answer: Insufficient data to solve accurately. Summary: - Solved problems 21, 23, 24, 25, 26 successfully. - Problem 22: solved for x but chord length not found. - Problems 27, 28: no real or consistent solution with given data.