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| ==== 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=]] |
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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]] | + | [[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]] | + | [[Image:KOER%20Circles_html_m5d49d71b.gif|link=]] |
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| ==== 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]] | + | [[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]] | + | [[Image:KOER%20Circles_html_m244a7f98.png|link=]] |
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| ==== 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]] | + | [[Image:KOER%20Circles_html_m6e667170.png|link=]] |
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| '''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 ? |
| + | ==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]] ==== |
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