"As nature puts forth its wonders in the garden, most people are oblivious to the massive calculations and mathematical work that have become so routine in nature."
Here Pappas wraps up the story about of the gardener who is surprised by all of the mathematics that appears in her garden. On this page she writes about a triple junction -- a point where three line segments meet and the angles at the intersection are each 120 degrees. According to Pappas, triple junctions are found in soap bubbles, as well as the cracking of earth or stone and on kernels on a cob of corn (bringing this back to the garden).
Lesson 15-6 of the U of Chicago text is on "Angles Formed by Tangents." The theorems in this lesson are similar to those in yesterday's lesson.
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Prove: Angle ABC = Arc AB/2
1. Draw diameter
2. Arc AD = 180 - AB 2. Arc Addition Postulate
3. CB perpendicular
4. Angle ABC = 90 - ABD 4. Angle Addition Postulate
5. Angle ABC = 180/2 - Arc AD/2 5. Inscribed Angle Theorem
6. Angle ABC = (180 - Arc AD)/2 6. Distributive Property
7. Angle ABC = Arc AB/2 7. Substitution Property of Equality
The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.
Given: Line AB secant, Ray EC tangent at point C, forming Angle E,
Arc AC = x, Arc BC = y
Prove: Angle E = (x - y)/2
Proof ("between a tangent and a secant"):
2. Angle DCA = x/2, EAC = y/2 2. Inscribed Angle Theorem
3. Angle DCA = EAC + E 3. Exterior Angle Theorem
4. Angle E = DCA - EAC 4. Subtraction Property of Equality
5. Angle E = x/2 - y/2 5. Substitution Property of Equality
6. Angle E = (x - y)/2 6. Distributive Property
In the text, the "between two tangents" is given as an exercise. The Given part of this proof with the way the points are labeled is completely different from the first part.
Given: Ray PV tangent at Q, Ray PU tangent at R
S on Circle O (same side of
Prove: Angle P = (Arc QTR - QSR)/2
Proof ("between two tangents"):
2. Angle VQR = Arc QTR/2, 2. Inscribed Angle Theorem
Angle PQR = Arc QSR/2
3. Angle VQR = PQR + P 3. Exterior Angle Theorem
4. Angle P = VQR - PQR 4. Subtraction Property of Equality
5. Angle P = Arc QTR/2 - QSR/2 5. Substitution Property of Equality
6. Angle P = (Arc QTR - QSR)/2 6. Distributive Property
In some ways, the Tangent-Chord Theorem is just like yesterday's Angle-Chord Theorem, except that one of the intercepted arcs is 0 degrees. The bonus question concerns a solar eclipse.