Question

In the adjoining figure, points $A,B,C,D$ lie on a circle. $AD=24$ and $BC=12$. What is the ratio of the area of $\triangle CBE$ to that of $\triangle ADE$? 

 

• A. $1:4$
• B. $1:2$
• C. $1:3$
• D. Insufficient data

Hint:

When two chords intersect inside a circle, triangles formed by corresponding chord-pairs are often similar; area ratios then follow from the square of the chord-length ratio.

Answer 1

The Correct Option is A

Let $E$ be the intersection of chords $BA$ and $CD$ (as in the figure). Claim: $\triangle ADE \sim \triangle CBE$.
Angles between intersecting chords inside a circle are equal when they intercept the same pair of arcs: \[ \angle AED = \angle CEB,\qquad \angle ADE = \angle CBE. \] Hence the triangles are similar with the correspondence \[ \triangle ADE \sim \triangle CBE\quad\Rightarrow\quad \frac{AD}{CB}=\frac{\text{scale of sides}}{}. \] Therefore, the ratio of their areas equals the square of the side ratio: \[ \frac{[CBE]}{[ADE]}=\left(\frac{CB}{AD}\right)^{\!2} =\left(\frac{12}{24}\right)^{\!2}=\frac{1}{4}. \]  Final Answer: \(\boxed{1:4}\) 

Answer 2

Instead of directly quoting that similar triangles' areas scale as the square of the similarity ratio, we can derive the area ratio using the formula \(\text{Area}=\tfrac12\times(\text{two sides})\times\sin(\text{included angle})\), and check each option.

1. Option A (\(1:4\)): Since \(E\) is the intersection of chords \(AB\) and \(CD\), \(\angle AED\) and \(\angle CEB\) are vertically opposite, hence equal; call this angle \(\theta\). Also, inscribed angles \(\angle ADE=\angle CBE\) (both subtend the same arc), so \(\triangle ADE\sim\triangle CBE\) with \(\dfrac{EA}{EB}=\dfrac{ED}{EC}=\dfrac{AD}{CB}=\dfrac{24}{12}=2\). The areas are \[ [ADE]=\tfrac12\cdot EA\cdot ED\cdot\sin\theta,\qquad [CBE]=\tfrac12\cdot EB\cdot EC\cdot\sin\theta, \] so \[ \frac{[CBE]}{[ADE]}=\frac{EB\cdot EC}{EA\cdot ED}=\left(\frac{CB}{AD}\right)^{2}=\left(\frac{12}{24}\right)^2=\frac14. \] This matches option A.
2. Option B (\(1:2\)): This would correspond to the sides ratio itself, not its square, so it is incorrect.
3. Option C (\(1:3\)): This does not match \(\left(\tfrac{12}{24}\right)^2=\tfrac14\), so it is incorrect.
4. Option D (insufficient data): Since the two chord lengths \(AD\) and \(BC\) are enough to fix the similarity ratio and hence the area ratio exactly, the data is sufficient, ruling out this option.

The trigonometric area formula confirms the ratio of the areas is \(1:4\).

Hence, the correct answer is option A: \(1:4\).