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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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Point V is in the exterior of circle Q (not shown) such that V Q ¯ is equal in length to the diameter of circle Q . Construct the two tangents to circle Q from point V . Then determine the measure of the angle that has vertex V and has the tangents as sides.

To determine

To construct:

Two tangent segments to a circle from an external point and determine the measure of the angle that has vertex as the external point and tangents as sides.

Explanation

The line that is perpendicular to the radius of a circle at its endpoint on the circle is a tangent to the circle. In the given figure, QT is a tangent to O at point T.

The measure of an inscribed angle of a circle is half the measure of its intercepted arc. Consider the figure below, let the inscribed angle be 2.

Then m2=12mHJ and generally, HKKJ.

The central angle formed by two radii,

It is given by m1=mAB and OA=OB.

The angle formed by radius drawn to tangent equals 90°.

Given:

The circle Q and an external point V.

Construction Strategy:

i) Draw a circle P with radius r.

ii) Take an external point E and draw EP¯ .

iii) Construct the perpendicular bisector of line EP¯. Start by placing the compass at E and adjusting the compass to any length <EP¯. Draw two arcs above and below the line from points E and P. Let the midpoint be M.

iv) With M as center and MP¯ (or ME¯) as the radius, construct a circle. Place ruler where the circles intersect, and draw the line segments T and V from point E. ET¯ or EV¯ is the desired tangent.

Construction:

Constructing a circle Q with radius r=2cm and an external point V. Joining QV¯=4cm.

Constructing the perpendicular bisector of line QV¯. Start by placing the compass at Q and adjusting the compass to any length <QV¯. Draw two arcs above and below the line from points Q and V. Let the midpoint be M.

Now with M as center and MQ¯ (or MV¯) as the radius, construct a circle. Place ruler where the circles intersect, and draw the line segments T and S from point V. VT¯ and SV¯ are the desired tangent

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