Chapter 9.7, Problem 24WE

(a)

To determine

To Find: the radius of circle drawn through the points X,Y and Z shown in figure.

Answer to Problem 24WE

  $5\sqrt{5}$

Explanation of Solution

Given:

Figure is shown below.

  

Calculation:

Extend YW as shown below.

  

A theorem state that “When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord”.

  $\begin{array}{l}XW\times WZ=YW\times WV\\ 8\times 12=6\times n\\ n=\frac{96}{6}\\ n=16\end{array}$

Mid-point of XZ and YV are $10\text{ and 11}$ . Draw lines from mid-points intersection at O as shown below.

  

From above figure, calculate radius OZ in triangle FOZ.

  $\begin{array}{l}O{Z}^2=O{F}^2+F{Z}^2\\ O{Z}^2={\left(11-6\right)}^2+{\left(20-10\right)}^2\\ O{Z}^2=25+100\\ O{Z}^2=125\\ r=OZ=5\sqrt{5}\end{array}$

Circle is drawn as shown below.

  

Therefore, the answer is $5\sqrt{5}$ .

(b)

To determine

To Find: the distance of centre from W in the figure shown.

Answer to Problem 24WE

  $\sqrt{29}$

Explanation of Solution

Given:

Figure is shown below.

  

Calculation:

Extend YW as shown below.

  

A theorem state that “When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord”.

  $\begin{array}{l}XW\times WZ=YW\times WV\\ 8\times 12=6\times n\\ n=\frac{96}{6}\\ n=16\end{array}$

Mid-point of XZ and YV are $10\text{ and 11}$ . Draw lines from mid-points intersection at O as shown below.

  

From above figure, calculate distance of centre O from W.

  $\begin{array}{l}O{W}^2=O{F}^2+W{F}^2\\ O{W}^2={\left(11-6\right)}^2+{\left(10-8\right)}^2\\ O{W}^2=25+4\\ O{W}^2=29\\ O{W}^2=\sqrt{29}\end{array}$

Therefore, the answer is $\sqrt{29}$ .