Circular Window Chord, Inscribed Angles, and Tangent Strip
A stained-glass studio is designing a circular window of radius 3.0 ft. A straight brass bar will be installed as chord AB, with AB = 4.2 ft. A triangular pane has its third vertex C on the major arc AB, and a decorative tangent strip will meet the circle at A. The studio's draft says angle ACB is 45° because an inscribed angle is half of a right angle, and that the tangent strip at A makes a 45° angle with chord AB. A second small pane uses point D on the minor arc AB instead of C. The fabricator needs the actual angle measures and the length of the minor arc AB to cut the brass bar and glass. The window frame is already fixed, so the chord cannot be moved; only the labels and cuts can be corrected.
1. Find the measure of angle ACB when C is on the major arc AB, and find the length of the minor arc AB. Use radius 3.0 ft and chord AB = 4.2 ft.
Read the worked answer
Let θ be the central angle subtending chord AB. The chord formula gives
AB = 2r sin(θ/2) 4.2 = 6 sin(θ/2) sin(θ/2) = 0.7 θ/2 ≈ 44.427° θ ≈ 88.854°.
So the minor arc AB measures about 88.85°. Point C is on the major arc, so angle ACB intercepts the minor arc and is half its measure:
m∠ACB ≈ 88.854°/2 ≈ 44.43°.
The draft's 45° is close but not exact. A 45° inscribed angle would intercept a 90° arc, which would require chord 2(3) sin 45° ≈ 4.243 ft, not 4.2 ft.
The minor arc length is s = rθ with θ in radians: s = 3(88.854°)(π/180°) ≈ 3(1.5508) ≈ 4.65 ft.
2. Find the measure of angle ADB when D is on the minor arc AB. Explain why it is not equal to angle ACB.
Read the worked answer
Angle ADB intercepts the major arc AB, whose measure is 360° − 88.854° = 271.146°. Therefore
m∠ADB = 271.146°/2 ≈ 135.57°.
It is not equal to angle ACB because the two points C and D lie on opposite arcs cut off by chord AB. An inscribed angle equals half the arc it intercepts, and the two arcs are complementary around the circle. The two angles are supplementary: 44.43° + 135.57° = 180°, as expected for opposite angles of a cyclic quadrilateral.
3. The draft says the tangent strip at A makes a 45° angle with chord AB. Using the tangent-chord relationship, what should the acute angle be, and why is 45° incorrect?
Read the worked answer
The tangent-chord theorem says the angle between a tangent and a chord equals the inscribed angle in the alternate segment. For chord AB, the relevant inscribed angle is the one subtending the minor arc AB, which is about 44.43°. Thus the acute angle between the tangent at A and chord AB should be about 44.43°, and the obtuse angle between the same lines is 180° − 44.43° = 135.57°.
The 45° draft is off by about 0.57°. That small error matters because the glass and brass are cut to fit a fixed frame; the tangent line is determined by the circle and chord, not by rounding the inscribed angle to a convenient whole number.