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Q9.

Expert-verified
Geometry
Found in: Page 336
Geometry

Geometry

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

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Illustration

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.

a. The figure is, Square.

b. The value of OZ is, 52.

See the step by step solution

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 XZ and OY YZ.

OX and OY is tangent to a circle.

That is;

OX || YZOY || XZ .......(1)

Step 2. Concept used.

Then addition of Adjutants angle is 180°.

X + Y =180°90° + Y= 180° Y =90°

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,

OX = YZOY= XZ ......(2)

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

Then, it can be concluded that,

OX = OY =YZ =XY

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,

OZ² =OX² + XZ²OZ² =5² + 5²OZ² =50

Step 3. Simplify.

OZ2=50OZ=52

Therefore, the value of OZ is, OZ=52.

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