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Q9.

Expert-verified
Found in: Page 336

### Geometry

Book edition Student Edition
Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen
Pages 227 pages
ISBN 9780395977279

# Draw $\odot O$ with perpendicular radii $\overline{OX}$ and $\overline{OY}$. Draw tangents to the circle at X and $Y$.$\left(a\right).$ If the tangents meet at $Z,$ what kind of figure is $OXYZ?$ Explain.$\left(b\right).$ If $OX=5,$ find $OZ$.

a. The figure is, Square.

b. The value of $OZ$ is, $5\sqrt{2}$.

See the step by step solution

## Part a. Step 1. Given information.

Given:

The circle with a center $O$ with perpendicular segment $OX$ and segment $OY$.

The tangent meet at $Z$.

Now, figure drawn by the information is,

In quadrilateral $OXYZ,$ $OX\text{}\perp \text{}XZ\text{and}OY\text{}\perp \text{}YZ.$

$OX$ and $OY$ is tangent to a circle.

That is;

$\begin{array}{l}OX\text{}||\text{}YZ\\ OY\text{}||\text{}XZ\end{array}$ .......(1)

## Step 2. Concept used.

Then addition of Adjutants angle is $180°$.

$\begin{array}{l}\angle \text{}X\text{}+\text{}\angle Y\text{}=180°\\ \begin{array}{l}90°\text{}+\text{}\angle Y=\text{}180°\\ \angle Y\text{}=90°\end{array}\end{array}$

Similarly prove that, $\angle Z\text{}=90°$.

Then all angles of quadrilateral are $90°$ and opposite side are parallel.

## Step 3. Opposite side of rectangle are equal.

Thus,

$\begin{array}{l}OX\text{}=\text{}YZ\\ OY=\text{}XZ\text{}......\left(2\right)\end{array}$

But $YX=XY$ (Tangent to the circle to same point)

Then, it can be concluded that,

$\begin{array}{l}OX\text{}=\text{}OY\\ =YZ\\ =XY\end{array}$

Then, all side is equal and all angles are $90°$.

## Part b. Step 1. Given information.

The given value is,

$OX=5$

In the above figure this is a square then all side is same.

## Step 2. Use Pythagoras Theorem.

In triangle $OXZ,$

$\begin{array}{l}\begin{array}{l}OZ²\text{}=OX²\text{}+\text{}XZ²\\ OZ²\text{}=5²\text{}+\text{}5²\end{array}\\ OZ²\text{}=50\end{array}$

## Step 3. Simplify.

$\begin{array}{l}\sqrt{O{Z}^{2}}=\sqrt{50}\\ OZ=5\sqrt{2}\end{array}$

Therefore, the value of $OZ$ is, $OZ=5\sqrt{2}$.