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Revision as of 16:09, 5 November 2019

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→‎Circles Tangents Problems
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==== Touching circles ====
 
==== Touching circles ====
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[[Image:KOER%20Circles_html_m5edc23ab.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m5edc23ab.gif]]
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[[Image:KOER%20Circles_html_m5edc23ab.gif|link=]]
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[[Image:KOER%20Circles_html_m202ccc14.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m202ccc14.gif]]
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[[Image:KOER%20Circles_html_m202ccc14.gif|link=]]
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[[Image:KOER%20Circles_html_m5d49d71b.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m5d49d71b.gif]]
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[[Image:KOER%20Circles_html_m5d49d71b.gif|link=]]
    
==== Common tangents ====
 
==== Common tangents ====
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The centres of the circles lie on the same side of the common tangent.(dct)
 
The centres of the circles lie on the same side of the common tangent.(dct)
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[[Image:KOER%20Circles_html_m202ccc14.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m202ccc14.gif]]
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[[Image:KOER%20Circles_html_m202ccc14.gif|link=]]
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[[Image:KOER%20Circles_html_m244a7f98.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m244a7f98.png]]
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[[Image:KOER%20Circles_html_m244a7f98.png|link=]]
    
==== Transverse common tangents ====
 
==== Transverse common tangents ====
 
The centres of the circles lie on either side of the common tangent(tct)
 
The centres of the circles lie on either side of the common tangent(tct)
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[[Image:KOER%20Circles_html_m6e667170.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m6e667170.png]]
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[[Image:KOER%20Circles_html_m6e667170.png|link=]]
    
'''Evaluation'''
 
'''Evaluation'''
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4. How many number of tangents that can be drawn through a point which is inside the circle ?
 
4. How many number of tangents that can be drawn through a point which is inside the circle ?
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==Proofs and verification of properties of tangents==
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The correct use of reasoning is at the core of mathematics, especially in constructing proofs. Many statements, especially in geometry. Recall that a proof is made up of several mathematical statements, each of which is logically deduced from a previous statement in the proof, or from a theorem proved earlier, or an axiom, or the hypotheses. The main tool, we use in constructing a proof, is the process of deductive reasoning.
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We start the study of this chapter in deductive reasoning using several examples.
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we can verify the theorems by practical construction. And also by using GeoGebra tool.
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==Tangents to a circles:==
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*Tangent: line that intersects a circle in exactly one point, called the point of tangency
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*Radius from centre of circle to the point of tangency is always perpendicular to the tangent line. If
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*The radius is not perpendicular to the line, the line is not tangent to the circle.
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*Recall the Pythagorean Theorem:
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*Use the fact that a tangent line and the radius through that point of tangency are perpendicular to solve for a third value. Show how you can also use this fact to deduce whether or not a line is tangent to a specific circle.
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*Tangents from an external point are equal in length.
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==Types of tangents==
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*Recognise the difference between a secant and a tangent of a circle.
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*Construct a tangent to a circle at a given point on it.
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*Construct and verify that, the radius drawn at the point of contact is perpendicular to the tangent.
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*Construct tangents to a circle from an external point.
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*Recognise the properties of direct common tangents and the transverse common tangents.
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==Touching circles==
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Common tangents of two circles How many common tangents do two circles have. Informally draw all different cases, with 0, 1, 2, 3, 4 common tangents.
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For any two different circles, there are five possibilities regarding their common tangents:
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*One circle lies inside the other. They have no common tangents.
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*One circle touches the other from inside. There is one common tangent, located at this touching point.
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*The two circles intersect in two points. They have two common tangents, which lie symmetrically to the axis connecting the two centres.
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*The two circles touch each other from outside. They have three common tangents.
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*The two circles lie outside of each other. They have four common tangents.
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==== '''Construction of tangents''' ====
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*[[Image:KOER%20Circles_html_50027288.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_50027288.png]]
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*<u>To draw a tangent to a circle from an external point </u>  [[Image:KOER%20Circles_html_m520802ec.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m520802ec.png]]
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*<u>To draw direct common tangents to two given circles of equal radii, with centres ‘d’ units apart. </u>  [[Image:KOER%20Circles_html_4b7743eb.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_4b7743eb.png]]
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*<u>To draw a direct common tangent to two circles of different radii. </u>  [[Image:KOER%20Circles_html_3b9c6f9.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_3b9c6f9.png]]To construct Transverse common tangents to two circles.
 
==== [[Circles Tangents Problems]] ====
 
==== [[Circles Tangents Problems]] ====
  
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