Can this Secant-Tangent Equation be Simplified Further?

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Homework Help Overview

The discussion revolves around a secant-tangent equation involving lengths related to a geometric configuration. Participants are exploring whether the equation can be simplified further, particularly focusing on the relationship between the lengths AD and AC.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the original equation and the steps taken to manipulate it, including cross-multiplication. There are questions about the validity of the steps and whether there is sufficient information to solve for AC. Some participants suggest alternative methods involving trigonometric relationships and the Pythagorean Theorem.

Discussion Status

The discussion is active, with participants questioning the adequacy of the given information and the steps taken in the calculations. Some guidance has been offered regarding the use of trigonometric identities and geometric principles to potentially simplify the problem further.

Contextual Notes

There appears to be a lack of sufficient information to fully resolve the problem, as some participants note that the original poster may not have all the necessary data to simplify the equation effectively.

Cyclopse
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Homework Statement


Is it possible to simplify this even further?
28b4d3b.jpg


Homework Equations


This is a Secant-tangent equation.
Secant: AD = 18/radical 3 AC = ??

The Attempt at a Solution


I cross multiplied to get 18=324radical 3 (AC)
and in the end i got 0 = (18radical3) (AC) which makes no sense

so is there a different way to simplify that?
 
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Cyclopse said:

Homework Statement


Is it possible to simplify this even further?
28b4d3b.jpg


Homework Equations


This is a Secant-tangent equation.
Secant: AD = 18/radical 3 AC = ??

The Attempt at a Solution


I cross multiplied to get 18=324radical 3 (AC)
Are you trying to solve for AC?
Cyclopse said:
and in the end i got 0 = (18radical3) (AC) which makes no sense
When you divide both sides by 18, you will be left with 1 on the left side, not 0.

If you are trying to find AC, divide both sides by 324√3.
Cyclopse said:
so is there a different way to simplify that?
 
Mark44 said:
Are you trying to solve for AC?

Yes.
Here is the original question
dh09i.jpg
 
Last edited:
It doesn't seem to me that you have enough information.

How did you go from AB = 18 and AD = 18/√3 to 18 = 324√3 * AC?
 
Mark44 said:
It doesn't seem to me that you have enough information.

How did you go from AB = 18 and AD = 18/√3 to 18 = 324√3 * AC?

I'm really terrible at maths...that's why I'm here.
 
Draw a radius from O to B, and call its length r. The radius OB is perpendicular to AB, so that ABO is a right triangle.

Let θ = the angle at A.

Then, from your given information,

cos(θ) = 18/(r + 18/√3), and
sin(θ) = r/(r + 18/√3)

Now you have two equations in two unknowns, so you should be able to solve for both unknowns, although you only need r. Once you know r, it's pretty easy to get AC.
 
Last edited:
Start out like Mark44 said by drawing a radius from O to B, and call its length r, but then apply the Pythagorean Theorem to the right triangle ABO to find r.

Chet
 
Chestermiller said:
Start out like Mark44 said by drawing a radius from O to B, and call its length r, but then apply the Pythagorean Theorem to the right triangle ABO to find r.
That's simpler than what I suggested. Simple is good!
 

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