The longest chord passes through the centre O, which is called the diameter of the circle.<\/span><\/p>\n <\/span><\/p>\n Given below are some of the important points or properties related to the chords of a circle.<\/span><\/p>\n <\/span><\/p>\n We can use 2 formulas to find the length of the circle\u2019s chord:<\/span><\/p>\n 1. Length of the chord using perpendicular distance from the centre<\/p>\n=2 \\times \\sqrt{\\left(r^{2}-d^{2}\\right)}<\/span>\n Proof:<\/b>\u00a0<\/span><\/p>\n In the circle given below, radius r is the hypotenuse of the triangle formed. Perpendicular bisector d will be one of the sides of the right-angled triangle.\u00a0<\/span><\/p>\n As per the property of chords, if the circle\u2019s radius is perpendicular to the chord, it bisects the chord.<\/span><\/p>\n Thus half of the chord forms the other side of the right-angled triangle.\u00a0<\/span><\/p>\n So, By Pythagoras theorem,<\/span><\/p>\n (1 \/ 2 \\text { chord })^{2}+d^{2}=r^{2}<\/span><\/span><\/p>\n Or, 1\/2 of Chord length =\\sqrt{\\left(r^{2}-d^{2}\\right)}<\/span><\/span><\/p>\n Therefore, chord length =2 \\times \\sqrt{\\left(r^{2}-d^{2}\\right)}<\/span><\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Here r is the circle\u2019s radius; d is the perpendicular distance from the chord to the circle\u2019s centre.<\/span><\/p>\n <\/p>\n 2. Chord length using trigonometry with angle\u00a0 = 2 \u00d7 r \u00d7 sin(\u03d5\/2)<\/p>\n Where \u03d5 is the angle subtended at the centre by the chord.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Theorem 1<\/b><\/p>\n Equal chords of the circle subtend equal angles at the centre of the circle.<\/span><\/b><\/p>\n <\/span><\/b><\/p>\n <\/b><\/p>\n Suppose AB and CD are two chords of the circle above with centre O and AB = CD. Then, \u2220AOB = \u2220COD.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Theorem 2<\/b><\/p>\n If the angles subtended at the centre by the chords are equal, the chords of the circle are equal.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n According to the theorem, If \u2220AOB = \u2220COD then, AB = CD.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Theorem 3\u00a0<\/b><\/p>\n A perpendicular from the centre of the circle to the chord divides the chord into two equal parts.<\/span><\/p>\n \u00a0<\/span><\/p>\n Theorem 4<\/b><\/p>\n If we draw a line through the centre of the circle to bisect the chord, it is perpendicular to the chord.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n If AM=BM, then \u2220OMA = \u2220OMB= 90\u00b0<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n Theorem 5<\/b><\/p>\n Angles subtended by a circle chord at different points on the same side of the circumference are of equal measurements.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Here, \u2220APB = \u2220AQB<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n The circle\u2019s radius is a line segment that joins the circle\u2019s centre to any point on the circle. On the other hand, the circle\u2019s chord is a line segment connecting any two points on the circumference of the circle. The circle\u2019s diameter is also a chord that passes through the centre and is equal to two times the radius length. It is also the longest chord of a circle.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/strong><\/span><\/p>\n 1. In the circle given below with O as the centre of the circle, the length of chord DC is 16 cm. Find the length of DE if OF is the circle\u2019s radius?<\/strong> <\/span><\/p>\n We know that the radius is perpendicular to the chord of the circle and is a perpendicular bisector. Therefore,\u00a0<\/span><\/p>\n DE = (\u00bd) \u00d7 AC\u00a0<\/span><\/p>\n \u00a0 \u00a0 \u00a0 = (\u00bd) \u00d7 16<\/span><\/p>\n \u00a0 \u00a0 \u00a0 = 8 cm<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n 2. In the figure given below, O is the centre of the circle with a radius of 5 cm. Find the length of chord DC if the perpendicular from the centre is 4 cm in length?<\/strong> <\/span><\/p>\n <\/span><\/p>\n Since OE is perpendicular to DC, so \u25b3DOE will be a right-angled triangle.\u00a0<\/span><\/p>\n In \u25b3DOE, from Pythagoras theorem,\u00a0<\/span><\/p>\n \\mathrm{OD}^{2}=\\mathrm{OE}^{2}+\\mathrm{DE}^{2}<\/span><\/span><\/p>\n Or \\mathrm{DE}^{2}=\\mathrm{OD}^{2}-\\mathrm{OE}^{2}<\/span><\/p>\n Substituting the values,\u00a0<\/span><\/p>\n \\mathrm{DE}^{2}=5^{2}-4^{2}<\/span><\/span><\/p>\n Thus, DE = \u221a9 = 3.\u00a0<\/span><\/p>\n <\/span><\/p>\n Therefore, Chord DC = 2 x 3 = 6 cm<\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Row for space” _builder_version=”4.10.6″ _module_preset=”default” locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.11″ _module_preset=”default” global_colors_info=”{}”][et_pb_divider show_divider=”off” _builder_version=”4.10.4″ _module_preset=”default” global_colors_info=”{}”][\/et_pb_divider][\/et_pb_column][\/et_pb_row][\/et_pb_section][et_pb_section fb_built=”1″ admin_label=”banner and faq Section” module_class=”mainsec2″ _builder_version=”4.10.4″ _module_preset=”default” custom_padding=”40px||0px||false|false” global_colors_info=”{}”][et_pb_row use_custom_gutter=”on” gutter_width=”1″ make_equal=”on” disabled_on=”on|on|off” admin_label=”banner Row” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fff7d6″ width=”100%” max_width=”1310px” height=”134px” custom_margin=”||50px||false|false” custom_padding=”12px||12px||true|false” border_radii=”on|11px|11px|11px|11px” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_image src=”https:\/\/stgwebsite.mindspark.in\/wordpress\/wp-content\/uploads\/2021\/08\/calloutImage.png” title_text=”calloutImage” show_bottom_space=”off” admin_label=”Image” module_class=”img1″ _builder_version=”4.10.8″ _module_preset=”default” width=”25px” height=”60px” custom_padding=”2px||2px||true|false” global_colors_info=”{}”][\/et_pb_image][et_pb_text module_class=”ftstyle” _builder_version=”4.9.10″ _module_preset=”default” text_orientation=”center” global_colors_info=”{}”]<\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/circle-1-300x253.png\")
<\/span><\/p>\n<\/b><\/h2>\n
Chords of a Circle: Important Points<\/b><\/h2>\n
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Chords of a Circle: Formula<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/circle-2-300x278.png\")
<\/span><\/p>\nChords of a Circle: Theorems<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Chord-of-a-Circle-01.png\")
<\/p>\n<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Chord-of-a-Circle-02.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Chord-of-a-Circle-03.png\")
<\/span><\/p>\nDifference Between Radius and Chords of a Circle<\/b><\/h2>\n
Examples<\/b><\/h2>\n
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/crir-300x254.png\")
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<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/crc2-300x269.png\")
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\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n