Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

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Short Answer

Draw $⊙ O$ with perpendicular radii $\bar{O X}$ and $\bar{O Y}$ . Draw tangents to the circle at X and $Y$ .
$(a) .$ If the tangents meet at $Z,$ what kind of figure is $O X Y Z ?$ Explain.
$(b) .$ If $O X = 5,$ find $O Z$ .

a. The figure is, Square.

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part a. Step 1. Given information.

Given:

The circle with a center $O$ with perpendicular segment $O X$ and segment $O Y$ .

The tangent meet at $Z$ .

Now, figure drawn by the information is,

In quadrilateral $O X Y Z,$ $O X ⊥ X Z and O Y ⊥ Y Z .$

$O X$ and $O Y$ is tangent to a circle.

That is;

$\begin{array}{l} O X | | Y Z \\ O Y | | X Z \end{array}$ .......(1)

Step 2. Concept used.

Then addition of Adjutants angle is $180 °$ .

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

Similarly prove that, $∠ Z = 90 °$ .

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

Therefore, this quadrilateral is Rectangle.

Step 3. Opposite side of rectangle are equal.

Thus,

$\begin{array}{l} O X = Y Z \\ O Y = X Z ...... (2) \end{array}$

But $Y X = X Y$ (Tangent to the circle to same point)

Then, it can be concluded that,

$\begin{array}{l} O X = O Y \\ = Y Z \\ = X Y \end{array}$

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

Part b. Step 1. Given information.

The given value is,

$O X = 5$

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

Step 2. Use Pythagoras Theorem.

In triangle $O X Z,$

\begin{array}{l} \begin{array}{l} O Z ² = O X ² + X Z ² \\ O Z ² = 5 ² + 5 ² \end{array} \\ O Z ² = 50 \end{array}

Step 3. Simplify.

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

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

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Geometry

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Short Answer

Draw $⊙ O$ with perpendicular radii $\bar{O X}$ and $\bar{O Y}$ . Draw tangents to the circle at X and $Y$ .
$(a) .$ If the tangents meet at $Z,$ what kind of figure is $O X Y Z ?$ Explain.
$(b) .$ If $O X = 5,$ find $O Z$ .

Step by Step Solution

Part a. Step 1. Given information.

Step 2. Concept used.

Step 3. Opposite side of rectangle are equal.

Part b. Step 1. Given information.

Step 2. Use Pythagoras Theorem.

Step 3. Simplify.

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Geometry

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Short Answer

Draw ⊙O with perpendicular radii OX¯ and OY¯. Draw tangents to the circle at X and Y.a. If the tangents meet at Z, what kind of figure is OXYZ? Explain.b. If OX=5, find OZ.

Step by Step Solution

Part a. Step 1. Given information.

Step 2. Concept used.

Step 3. Opposite side of rectangle are equal.

Part b. Step 1. Given information.

Step 2. Use Pythagoras Theorem.

Step 3. Simplify.

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Draw $⊙ O$ with perpendicular radii $\bar{O X}$ and $\bar{O Y}$ . Draw tangents to the circle at X and $Y$ .
$(a) .$ If the tangents meet at $Z,$ what kind of figure is $O X Y Z ?$ Explain.
$(b) .$ If $O X = 5,$ find $O Z$ .