Mathematics Asked on January 1, 2022

A diameter $AB$ and a chord $CD$ of a circle $k$ intersect at $M.$ $CE$ and $DF$ are perpendiculars from $C$ and $D$ to $AB$. $(A,E,M,F,B$ lie on AB in that order$)$. What is the length of $CD$ if $AE=1,FB=49$ and $MC:MD=2:7$?

How do I approach the given problem? I would be very grateful if you could give me some hints and tips to follow. I see that the triangles $CEM$ and $DFM$ are similar and $dfrac{MC}{MD}=dfrac{CE}{DF}=dfrac{EM}{FM}=dfrac{2}{7}.$

Let $CM=2x$ and $EM=2y$.

Thus, $$MF=7y,$$ $$MD=7x$$ and since $measuredangle ACB=90^{circ}$, we obtain $$CE^2=AEcdot EB.$$ Also, $$AMcdot MB=CMcdot MD$$ and we obtain the following system: $$(2x)^2-(2y)^2=1cdot(2y+7y+49)$$ and $$(1+2y)(7y+49)=2xcdot7x.$$ The last equality it's $$(1+2y)(y+7)=2x^2$$ and we can substitute $2x^2$ in the first equation.

Thus $$2(1+2y)(y+7)-4y^2=9y+49.$$ Can you end it now?

I got $CD=39$.

Answered by Michael Rozenberg on January 1, 2022

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP