9. $AB$ is parallel to $EC$ and $AB=(1+3 \sqrt{5}) cm$. $E$ is a point on $AD$ such that $AE:ED=\sqrt{5}:3$. Find
(a) $\frac{EC}{AB}$ in the form of $p+q \sqrt{5}$, where $p$ and $q$ are rational numbers.
Answer (1)
Use similarity of triangles to establish the ratio A B EC = A E D E .
Find the reciprocal of the given ratio E D A E = 3 5 to get A E D E = 5 3 .
Rationalize the denominator of 5 3 by multiplying the numerator and denominator by 5 , resulting in 5 3 5 .
Express the result in the form p + q 5 , where p = 0 and q = 5 3 , so the final answer is 5 3 5 .
Explanation
Problem Analysis and Given Information We are given that A B is parallel to EC , and E lies on A D . We are also given that $AB = (1 + 3
\sqrt{5}) cm an d AE:ED = \sqrt{5}:3 . O u r g o a l i s t o f in d t h er a t i o \frac{EC}{AB} in t h e f or m p + q\sqrt{5} , w h ere p an d q$ are rational numbers.
Using Similarity and Ratios Since A B ∥ EC , we can say that △ A BE ∼ △ C D E . Therefore, the ratio of corresponding sides is equal, i.e., A B EC = A E D E We are given that E D A E = 3 5 . Taking the reciprocal of this ratio, we get A E E D = 5 3 Thus, A B EC = 5 3 To express this in the form p + q 5 , we need to rationalize the denominator.
Rationalizing the Denominator and Expressing the Result To rationalize the denominator, we multiply both the numerator and the denominator by 5 :
A B EC = 5 3 × 5 5 = 5 3 5 Now we can express this in the form p + q 5 , where p and q are rational numbers. In this case, p = 0 and q = 5 3 .
Therefore, A B EC = 0 + 5 3 5 .
Final Answer Thus, the ratio A B EC is 5 3 5 , which can be written as 0 + 5 3 5 .
Examples
Understanding ratios and proportions is crucial in various real-life scenarios. For instance, when scaling recipes, you need to maintain the correct ratios of ingredients to ensure the dish tastes as expected. Similarly, in architecture and engineering, maintaining precise ratios is essential for creating stable and aesthetically pleasing structures. In finance, understanding ratios helps in analyzing a company's performance and making informed investment decisions. The ability to work with ratios and proportions is a fundamental skill that applies across many disciplines.
