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From Karnataka Open Educational Resources
→Circles Tangents Problems
==== Touching circles ====
==== Touching circles ====
[[Image:KOER%20Circles_html_m5edc23ab.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m5edc23ab.gif]]
[[Image:KOER%20Circles_html_m5edc23ab.gif|link=]]
[[Image:KOER%20Circles_html_m202ccc14.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m202ccc14.gif]]
[[Image:KOER%20Circles_html_m202ccc14.gif|link=]]
[[Image:KOER%20Circles_html_m5d49d71b.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m5d49d71b.gif]]
[[Image:KOER%20Circles_html_m5d49d71b.gif|link=]]
==== Common tangents ====
==== Common tangents ====
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)
[[Image:KOER%20Circles_html_m202ccc14.gif|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m202ccc14.gif]]
[[Image:KOER%20Circles_html_m202ccc14.gif|link=]]
[[Image:KOER%20Circles_html_m244a7f98.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m244a7f98.png]]
[[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)
[[Image:KOER%20Circles_html_m6e667170.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_m6e667170.png]]
[[Image:KOER%20Circles_html_m6e667170.png|link=]]
'''Evaluation'''
'''Evaluation'''
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 ?
==Proofs and verification of properties of tangents==
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.
We start the study of this chapter in deductive reasoning using several examples.
we can verify the theorems by practical construction. And also by using GeoGebra tool.
==Tangents to a circles:==
*Tangent: line that intersects a circle in exactly one point, called the point of tangency
*Radius from centre of circle to the point of tangency is always perpendicular to the tangent line. If
*The radius is not perpendicular to the line, the line is not tangent to the circle.
*Recall the Pythagorean Theorem:
*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.
*Tangents from an external point are equal in length.
==Types of tangents==
*Recognise the difference between a secant and a tangent of a circle.
*Construct a tangent to a circle at a given point on it.
*Construct and verify that, the radius drawn at the point of contact is perpendicular to the tangent.
*Construct tangents to a circle from an external point.
*Recognise the properties of direct common tangents and the transverse common tangents.
==Touching circles==
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.
For any two different circles, there are five possibilities regarding their common tangents:
*One circle lies inside the other. They have no common tangents.
*One circle touches the other from inside. There is one common tangent, located at this touching point.
*The two circles intersect in two points. They have two common tangents, which lie symmetrically to the axis connecting the two centres.
*The two circles touch each other from outside. They have three common tangents.
*The two circles lie outside of each other. They have four common tangents.
==== '''Construction of tangents''' ====
*[[Image:KOER%20Circles_html_50027288.png|link=https://karnatakaeducation.org.in/KOER/en/index.php/File:KOER_Circles_html_50027288.png]]
*<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]]
*<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]]
*<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]] ====
