Question

In a circle, chord AB and chord CD intersect at E such that AE : EB = 2: 3 and CE: ED = 5: 2. If AB =X and CD = y, which of the following is true?

• A. $$\frac{x^2}{y^2}=\frac{15}{16}$$
• B. $$\frac{x}{y}=\frac{6}{15}$$
• C. $$\frac{x}{y}=\frac{10}{9}$$
• D. $$\frac{x^2}{y^2}=\frac{125}{147}$$

Answer 1

Answer:

$$\frac{x^2}{y^2}=\frac{125}{147}$$

Using the intersecting chords theorem:

$AE\times EB=CE\times ED$

Given:

$AE:EB=2:3,\qquad CE:ED=5:2$

Let:

$AE=2a,\ EB=3a,\ CE=5b,\ ED=2b$

Then:

$(2a)(3a)=(5b)(2b)$
$6a^2=10b^2$

$\frac{a^2}{b^2}=\frac53$

Now,

$X=AB=2a+3a=5a$

and

$Y=CD=5b+2b=7b$

Therefore,

$\frac{X^2}{Y^2}$

$=\frac{25a^2}{49b^2}$

$=\frac{25}{49}\times\frac53$
$=\boxed{\frac{125}{147}}$