In figure 3.94, (1) m(arc CE) = 54°, m(arc BD) = 23°, find measure of ∠CAE. (2) If AB = 4.2, BC = 5.4, AE = 12.0, find AD (3) If AB = 3.6, AC = 9.0, AD = 5.4, find AE

In figure 3.94,

(1) m(arc CE) = \(54^\circ\), m(arc BD) = \(23^\circ\), find measure of \(\angle CAE\).

(2) If AB = 4.2, BC = 5.4, AE = 12.0, find AD

(3) If AB = 3.6, AC = 9.0, AD = 5.4, find AE

Figure: A is outside the circle; two secants A–B–C and A–D–E are drawn from A.

Doubtly Asked 24 Oct 2026 Save WhatsApp Report

1 answer

Answer: (1) \(15.5^\circ\) (2) AD = 3.36 (3) AE = 6

(1) Angle between two secants meeting outside the circle:

\[\angle CAE = \frac{1}{2}[m(\text{arc CE}) - m(\text{arc BD})]\] \[= \frac{1}{2}(54^\circ - 23^\circ) = 15.5^\circ\]

(2) AC = 4.2 + 5.4 = 9.6. By the theorem of secants intersecting outside a circle, \(AB \times AC = AD \times AE\):

\[4.2 \times 9.6 = AD \times 12\] \[AD = \frac{40.32}{12} = 3.36\]

(3)

\[3.6 \times 9 = 5.4 \times AE\] \[AE = \frac{32.4}{5.4} = 6\]

Doubtly AI · 24 Oct 2026 AI-verified · Oct 2026 Report

Was this answer helpful?