<\/span><\/p>\n <\/span><\/p>\n In <\/span>\u2206<\/span>ABC, the altitudes are AE, BF and CD. The orthocentre of the triangle is point O.<\/span><\/p>\n <\/p>\n The orthocentre of a triangle can be constructed by following the steps given below:<\/span><\/p>\n \n The properties of the orthocentre depend on the type of triangle. For some triangles, the orthocentre may even lie outside the triangle. The properties of orthocentre are:<\/span><\/p>\n \n The triangle ABC is an acute triangle, it can be seen that the orthocentre lies inside the triangle.<\/p>\n \n <\/span><\/p>\n 2. The orthocentre of an obtuse triangle, lies outside the triangle.<\/span><\/p>\n \n Triangle DEF is an obtuse triangle, hence its orthocentre lies outside the triangle.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n \n For the triangle PQR, the orthocentre is the vertex R.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n The orthocentre formula can help in determining the coordinates of the orthocentre of any triangle. Let\u2019s derive the formula for the orthocentre by taking a triangle ABC.<\/span><\/p>\n As shown in the figure of <\/span>\u2206<\/span>ABC<\/span><\/p>\n Let the coordinates of the vertices be \\mathrm{A}\\left(x_{1}, y_{1}\\right), \\mathrm{B}\\left(x_{2}, y_{2}\\right), \\mathrm{C}\\left(x_{3}, y_{3}\\right), \\mathrm{O}(x, y)<\/span>.<\/span><\/p>\n From the figure, AE, BF and CD are the altitudes of the triangle. O is the orthocentre.<\/span><\/p>\n Step 1:<\/b> First calculate the slope of any two sides of the triangle using the formula<\/span><\/p>\n \\text { Slope }(\\mathrm{m})=\\frac{y_{2}-y_{1}}{x_{2}-x_{1}}<\/span><\/span><\/p>\n [ This is the formula for calculating the slope of the line using two points on the line\\left.\\left(x_{1}, y_{1}\\right) \\text { and }\\left(x_{2}, y_{2}\\right)\\right]<\/span><\/span><\/p>\n The slope of the side BC and AC can be calculated using this formula.<\/span><\/p>\n \\text{The slope of }\\mathrm{BC}, m_{B C}=\\frac{y_{3}-y_{2}}{x_{3}-x_{2}}<\/span><\/span><\/p>\n <\/span><\/p>\n Step 2<\/b>: The perpendicular slope of a line is given by the formula,<\/span><\/p>\n The perpendicular slope of line =-\\frac{1}{m} .<\/span><\/span><\/p>\n Using this formula, the slope of perpendiculars to BC and AC can be found out.<\/span><\/p>\n The slope of the perpendiculars to BC and AC will actually be the slope of altitudes, AE and BF respectively.<\/span><\/p>\n <\/span><\/p>\n \\text{The slope of } \\mathrm{AE}, m_{A E}=-\\frac{1}{m_{B C}}<\/span><\/span><\/p>\n <\/span><\/p>\n Step 3<\/b>: Calculating the slope of the altitudes AE and BF using the point-slope formula:<\/span><\/p>\n \\text{The slope of } \\mathrm{AE}, m_{A E}=\\frac{y-y_{1}}{x-x_{1}}<\/span><\/span><\/p>\n Thus, solving the two equations for the given values will give the coordinates of the orthocentre O(x, y).<\/span><\/p>\n <\/span><\/p>\n Examples 1: <\/strong>Name the vertices, altitudes and orthocentre of the given triangle.<\/span><\/p>\n Solution:<\/strong><\/p>\n In the above figure for\u00a0<\/span>\u25b3<\/span>ABC,<\/span><\/p>\n The Vertices of the triangle are A, B, and C.<\/span><\/p>\n Sides of the <\/span>\u2206<\/span> are AB, BC, AC and the Altitudes are AE, BF, CD.<\/span><\/p>\n The Orthocentre of the triangle is O.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Example 2<\/strong>:<\/strong> Determine the coordinates of the orthocentre of the triangle whose vertices are: A(4, 8), B(-2, 0), C(2, 4).<\/span><\/p>\n Solution:<\/strong><\/p>\n Let us calculate the slope of any two sides of the triangle.<\/span><\/p>\n The slope of the side BC and AC can be calculated using the point-slope formula.<\/span><\/p>\n \\text { Slope }(\\mathrm{m})=\\frac{y_{2}-y_{1}}{x_{2}-x_{1}}<\/span><\/span><\/p>\n The slope of \\mathrm{BC}, m_{B C}=\\frac{y_{3}-y_{2}}{x_{3}-x_{2}}=\\frac{4-0}{2-(-2)}=\\frac{4}{4}=1<\/span><\/span><\/p>\n <\/span><\/p>\n The slope of \\text { AC, } m_{A C}=\\frac{y_{3}-y_{1}}{x_{3}-x_{1}}=\\frac{4-8}{2-4}=\\frac{-4}{-2}=2<\/span><\/span><\/p>\n Now, the perpendicular slopes of the sides BC and AC.<\/span><\/p>\n <\/span><\/p>\n The perpendicular slope of side BC =-\\frac{1}{m_{B C}}=-\\frac{1}{1}=-1 \\ldots(1)<\/span><\/span><\/p>\n <\/span><\/p>\n The perpendicular Slope of side AC =-\\frac{1}{m_{A C}}=-\\frac{1}{2} \\cdots(2)<\/span><\/span><\/p>\n <\/span><\/p>\n Let the coordinates of the Orthocentre be (x, y).<\/span><\/p>\n Now by point-slope form, the slope of the line passing through vertex A(4, 8) and orthocentre (x, y) is:<\/span><\/p>\n The slope of the perpendicular from vertex \\mathrm{A}=\\frac{y-8}{x-4}<\/span><\/span><\/p>\n From (2) we have the slope of the perpendicular from A=-\\frac{1}{2}<\/span><\/span><\/p>\n \\Rightarrow \\frac{y-8}{x-4}=-\\frac{1}{2} <\/span><\/span><\/p>\n <\/p>\n Similarly, the slope of line passing through the vertex B(-2, 0) and orthocentre (x, y) is:<\/span><\/p>\n Slope of perpendicular from vertex B=\\frac{y-0}{x-(-2)}=\\frac{y}{x+2}<\/span><\/span><\/p>\n From (1) we have slope of perpendicular from B = -1\u00a0<\/span><\/p>\n <\/span><\/p>\n Solving equations (3) and (4)<\/span><\/p>\n x + 2y = 20<\/span><\/p>\n x + \u00a0 y = -2\u00a0<\/span><\/p>\n – \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 +<\/span><\/p>\n —————–<\/span><\/p>\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y = 22<\/span><\/p>\n <\/span><\/p>\n \u2234<\/span> x = 20 – 2y = 20 \u2013 2(22) = 20 \u2013 44 = -24<\/span><\/p>\n <\/span><\/p>\n Hence the orthocentre of the given triangle is at point (-24, 20).<\/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![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/iamge22-300x211.png\")
<\/span><\/p>\nConstruction of Orthocentre<\/b><\/h2>\n
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Properties\u00a0<\/b><\/h2>\n
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![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/ORTHOCENTRE-02-300x214.png\")
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Orthocentre Formula<\/b><\/h2>\n
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\\text{The\u00a0 slope of }\\mathrm{AC},m_{A C}=\\frac{y_{3}-y_{1}}{x_{3}-x_{1}}<\/span><\/span><\/p>\n
\\text{The slope of } \\mathrm{BF}, m_{B F}=-\\frac{1}{m_{A C}}<\/span><\/span><\/p>\n
\\text{The slope of } \\mathrm{BF}, m_{B F}=\\frac{y-y_{2}}{x-x_{2}}<\/span><\/span><\/p>\nExamples<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/sssss-300x228.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/sss222-300x259.png\")
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\\Rightarrow 2(y-8)=-(x-4) <\/span><\/span><\/p>\n
\\Rightarrow x+2 y=4+16=20 \\ldots(3)<\/span><\/span><\/p>\n
\\Rightarrow \\frac{y}{x+2}=-1 <\/span><\/span><\/p>\n
\\Rightarrow \\mathrm{y}=-(\\mathrm{x}+2) <\/span><\/span><\/p>\n
\\Rightarrow \\mathrm{x}+\\mathrm{y}=-2 \\ldots(4)<\/span>
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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