Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Power of a Point Theorem/Introductory Problem 3"

 
(No difference)

Revision as of 06:40, 1 July 2006

Problem

(ARML) In a circle, chords $AB$ and $CD$ intersect at $R$. If $AR:BR = 1:4$ and $CR:DR = 4:9$, find the ratio $AB:CD.$

Popprob3.PNG

Solution

Letting $AR = x$ makes $BR = 4x$. Similarly, letting $CR = 4y$ makes $DR = 9y$. Thus $AB = AR + BR = x + 4x = 5x$ and $CD = CR + DR = 4y + 9y = 13y$. We therefore seek $\frac{AB}{CD} = \frac{5x}{13y}$.

From the Power of a Point Theorem we have that

$x\cdot 4x = 4y\cdot 9y\Rightarrow \left(\frac xy\right)^2 = 9$

which gives $\frac xy = \pm 3$ so we take $\frac xy = 3$.

Finally

$\frac{5x}{13y}=\frac 5{13}\cdot \frac xy = \frac 5{13}\cdot 3 = \frac{15}{13}.$

Back to the Power of a Point Theorem.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Power_of_a_Point_Theorem/Introductory_Problem_3&oldid=5736"