Answer:
2. arc AB = 112.63°; angle ADB = 56.315°
3. AE = 9
4. BC = 2√13 ≈ 7.21
Step-by-step explanation:
You want various arcs, angles, and segment measures in the circle diagram given.
2. Arc AB
Since AC is a diameter of the circle, arc AC will be 180°. Arc AB is the difference between that and arc BC:
Arc AB = 180° -67.37°
Arc AB = 112.63°
Angle ADB is an inscribed angle subtending arc AB, so the angle measure is half the arc measure:
Angle ADB = 1/2(112.63°)
Angle ADB = 56.315°
3. AE
The product of segments AE and EC is the same as the product of segments BE and ED:
AE(13 -AE) = 6·6
9·(13 -9) = 9·4 = 36
The length of segment AE is 9 units.
4. BC
The length of BC is the root of the product of CE and AC:
BC = √(4·13)
BC = 2√13 ≈ 7.21
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Additional comment
The quadratic equation in 3 can be solved a number of ways. We initially found the solution by graphing. It can also be found by considering factors of 36 that total 13. We know AE > CE, so AE is the greater of the two factors 4 and 9.
The relation used in 4 is a consequence of the similarity of triangles CBE, BAE, and CAB. This similarity means CE/CB = CB/CA, or CB² = CE·CA. You could also use the Pythagorean relation CB² = CE² +BE² to get the same result.
