Line 7: |
Line 7: |
| #The students have to prove thne angle PAQ=twise the angle OPQ. | | #The students have to prove thne angle PAQ=twise the angle OPQ. |
| ===Geogebra file=== | | ===Geogebra file=== |
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> | + | <span> </span> |
| + | |
| + | <span></span><div id="ggbContainer14a2b70cb23c6cb8374fd4f22e997e08"></div><span></span> |
| | | |
| ==Concepts used== | | ==Concepts used== |
Line 154: |
Line 156: |
| #OB and OF are the radii of the semicircle with center "O". | | #OB and OF are the radii of the semicircle with center "O". |
| #PC and PF are the radii of the circle with center "P". | | #PC and PF are the radii of the circle with center "P". |
| + | |
| + | ==Ex 4.4.2== |
| + | #Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length. |
| + | '''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ. |
| + | |
| + | '''TO PROVE :-''' AB = CD, |
| + | |
| + | '''CONSTRUCTION :-''' Join OA & OD. |
| + | |
| + | '''PROOF :-''' |
| + | {[Consider In ∆AOP & ∆DOQ OA = OD OP = OQ Angle APO = Angle DQO ∆AOP ≡ ∆DOQ AP = DQ Let AB = AP + BP = AP + AP = 2AP AB = 2DQ ---------- 1. and CD = CQ + DQ = DQ + DQ CD = 2DQ --------- 2. From equtn 1 & equtn 2 AB = CD |
| + | Radii of the circle Equi distances from circle |
| + | SAS Axiom |
| + | Acording to properties of SAS axiom. |
| + | |
| + | Perpendicular drawn from centre to chord which bisect the chord, i.e. AP = BP. |
| + | |
| + | Perpendicular drawn from centre to chord which bisect the chord, i.e. CQ = DQ Acording to AXIOM-1]} |
| + | |
| + | angle |
| + | {| class="wikitable" |
| + | |- |
| + | |'''Steps''' |
| + | |'''Explanation''' |
| + | |- |
| + | |[[Image:solution.png|300px|link=http://karnatakaeducation.org.in/KOER/en/index.php/File:Solution.png]] |
| + | |Explanation for thestep |
| + | |- angle |
| + | |Write the step |
| + | |Explanation for thestep |
| + | |}|} |