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ABC is a right triangle with ∠B=90∘, AB=5, BC=12, and AB tangent to a circle. CP and CQ are also tangent to the circle at P and Q.  Find the length of chord PQ.

 

 Jun 24, 2020
 #1
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ABC is a right triangle with \(\angle B=90^\circ\), AB=5, BC=12, and AB tangent to a circle. 

CP and CQ are also tangent to the circle at P and Q. 

Find the length of chord \(\color{red}PQ\).

 

\(\text{Let $PQ=x$} \\ \text{Let $PC=QC$} \\ \text{Let $PA=AS=y$} \\ \text{Let $AC=\sqrt{12^2+5^2}=13$} \)

 

\(\begin{array}{|lrcll|} \hline (1): & PC &=& QC \quad | \quad QC = 12+r,\ PC=y+AC,\ AC= \sqrt{12^2+5^2}= 13 \\ & y+13 &=& 12+r \\ & \mathbf{y} &=& \mathbf{r-1} \\\\ (2): & AS &=& AB - SB \quad | \quad AS=y,\ AB=5,\ SB = r \\ & y &=& 5-r \\ & r &=& 5-y \quad | \quad y=r-1\\ & r &=& 5-(r-1) \\ & r &=& 5-r+1 \\ & 2r &=& 6 \\ & \mathbf{r} &=& \mathbf{3} \\ \hline \end{array}\)

 

 

\(\begin{array}{|rcll|} \hline RC &=& \sqrt{(r+12)^2+r^2 } \quad | \quad \mathbf{r=3} \\ RC &=& \sqrt{(3+12)^2+3^2 } \\ RC &=& \sqrt{15^2+3^2 } \\ \mathbf{RC} &=& \mathbf{\sqrt{234 }} \\ \hline \cos(\varphi) &=& \dfrac{RQ}{RC} \quad | \quad RQ=r=3,\ \mathbf{RC=\sqrt{234 }} \\ \cos(\varphi) &=& \dfrac{3}{\sqrt{234 }} \\ && \boxed{\text{Formula:} \cos(2\varphi)= 2\cos^2(\varphi)-1 } \\ \cos(2\varphi) &=& 2\cos^2(\varphi)-1 \quad | \quad \cos(\varphi) = \dfrac{3}{\sqrt{234 }}\\ \cos(2\varphi)&=& 2*\left(\dfrac{3}{\sqrt{234 }} \right)^2-1 \\ \cos(2\varphi)&=& 2*\dfrac{9}{ 234 }-1 \\ \cos(2\varphi)&=& \dfrac{9}{ 117 } -1 \\ \mathbf{\cos(2\varphi)} &=& \mathbf{-\dfrac{108}{ 117 }} \\ \hline \end{array} \)

 

cos-rule:

\(\begin{array}{|rcll|} \hline \mathbf{\text{In $\triangle RPQ$ }:} \\ \hline x^2 &=& r^2+r^2 -2rr\cos(2\varphi) \\ x^2 &=& 3^2+3^2 -2*3*3*\left(-\dfrac{108}{ 117 }\right) \\ x^2 &=& 18\left(1+\dfrac{108}{ 117 }\right) \\ x^2 &=& \dfrac{18*225}{ 117 } \\ x^2 &=& \dfrac{225}{ 6.5 } \\ \mathbf{x} &=& \mathbf{\dfrac{15}{ \sqrt{6.5} }} \\ \mathbf{x} &=&\mathbf{ 5.88348405415} \\ \hline \end{array}\)

 

The length of chord PQ is \(\mathbf{\approx 5.9}\)

 

laugh

 Jun 24, 2020
edited by heureka  Jun 24, 2020

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