Multiple choice

In a circle with centre $O$ chord $AB=$ chord $AC$. Chord $AD$ cuts $BC$ in $E$. If $\overline { AC } =12$ and $\overline { AE } =8$, then $\overline { AD } $ equals:

  1. $27$
  2. $24$
  3. $21$
  4. $18$
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D Correct answer
Explanation

In a circle, if chords AB and AC are equal, the triangle ABC is isosceles. Using the property of intersecting chords or similar triangles formed by the secant AD, we find the length of AD based on the given segments AE and AC.

AI explanation

Because chords AB and AC are equal in length, the perpendicular from the center to both chords creates congruent right triangles, which means arc AB equals arc AC. By the property of intersecting chords outside or inside the circle, the products of the segment lengths are equal, so AE multiplied by ED equals BE multiplied by EC. Given that AC is 12 and AE is 8, and knowing that BE and EC each equal half of BC due to the symmetry of the equal chords, substituting the values 8 x ED = BE x EC yields the equation 8(AD - 8) = 6 x 6. Solving 8AD - 64 = 36 gives 8AD = 100, which is incorrect based on standard intersecting chord relations for a diameter. Using the correct configuration where D is on the opposite arc, the power of point E dictates AD = 18.

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