The main importance of altitude is in the calculation of the area of a triangle.<\/span><\/p>\n Area of Triangle<\/b> = \\frac{1}{2}<\/span> <\/b>\u00d7 Base \u00d7 Height<\/span><\/p>\n Here, height is the altitude of the triangle.<\/span><\/p>\n The point at which the three altitudes of a triangle meet(intersect) is the orthocenter of that triangle.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n In \u2206ABC the perpendicular drawn from vertex A to the side BC is the altitude of the triangle. Where BC is the base with respect to the altitude AD. D is the point of intersection of base and altitude.<\/span><\/p>\n <\/p>\n The various properties of altitudes are:<\/span><\/p>\n <\/p>\n <\/p>\n In an equilateral triangle, the altitude bisects its base and the angle at its respective vertex.<\/span><\/p>\n The image shows an equilateral triangle <\/span>\u2206<\/span>XYZ.<\/span><\/p>\n <\/span><\/p>\n We know, for an equilateral triangle all the angles are equal to 60\u00b0and let all sides of the triangle be \u2018A\u2019. Let \u2018H\u2019 be the altitude. In the triangle above we have: <\/p>\n\\sin 60^{\\circ}=\\frac{\\sqrt{3}}{2}<\/span>\n <\/p>\n\\therefore \\frac{H}{A}=\\frac{\\sqrt{3}}{2} \\Rightarrow H=\\frac{\\sqrt{3}}{2} \\times A \\text {. }<\/span>\n Therefore, for an equilateral triangle, its Altitude is \\frac{\\sqrt{3}}{2} \\times(\\text { side })<\/span>.<\/span><\/p>\n <\/p>\n We know, for an isosceles triangle two sides have equal dimensions.\u00a0<\/span><\/p>\n <\/span><\/p>\n \u2206<\/span>PQR is an isosceles triangle, PQ = PR = x, QR = y and let PO = h.<\/span><\/p>\n The altitude PO of the triangle bisects the side QR <\/span>\u21d2 <\/span>QO = OR\u00a0 = y\/2<\/span><\/p>\n \u2206<\/span>POQ is a right-angle triangle, hence, by Pythagoras Theorem,<\/span><\/p>\nP O^{2}+Q O^{2}=P Q^{2}<\/span>\n Substituting the values of PO, QO and PQ we get,\u00a0<\/span><\/p>\nh^{2}+\\left(\\frac{y}{2}\\right)^{2}=x^{2} \\Rightarrow h^{2}=x^{2}-\\left(\\frac{y}{2}\\right)^{2}<\/span>\n \\therefore h=\\sqrt{x^{2}-\\frac{y^{2}}{4}}<\/span><\/p>\n <\/p>\n <\/b><\/p>\n <\/span><\/p>\n The above triangle represents a right-angled triangle which is right-angled at A. A perpendicular is dropped from vertex A onto the hypotenuse BC.<\/span><\/p>\n AD is thus the altitude of the triangle. Let its length be h.<\/span><\/p>\n By the right triangle altitude theorem this altitude divides the triangle, <\/span>\u2206<\/span>ABC into two similar triangles, i.e., <\/span>\u2206<\/span>BDA<\/span> \u301c<\/span>\u2206<\/span>ACD.<\/span><\/p>\n As\u00a0 <\/span>\u2206<\/span>BDA<\/span> \u301c <\/span>\u2206<\/span>ACD, we have\u00a0<\/span><\/p>\n \\frac{B D}{A D}=\\frac{A D}{D C}<\/span><\/span><\/p>\n \\Rightarrow A D^{2}=B D \\times D C<\/span><\/span><\/p>\n \\therefore h^{2}=p \\times q<\/span><\/span><\/p>\n h=\\sqrt{p q}<\/span><\/span><\/p>\n <\/span><\/p>\n AB and AC are also altitudes of the triangle shown above and as they are perpendicular to each other when one is considered to be the altitude the other will be the base.<\/span><\/p>\n We know, that in a triangle when one angle is greater than 90<\/span>\u00b0<\/span>, it is termed as an obtuse triangle. In the case of an obtuse triangle, the altitude lies outside the triangle and is constructed by extending the base as seen in the figure below.<\/span><\/b><\/p>\n <\/span><\/b><\/p>\n Example 1<\/strong>:<\/strong> Calculate the length of altitude of an equilateral triangle in which each side is 12 units in length. Also, determine the area of the same triangle.<\/span><\/p>\n Solution:<\/strong><\/p>\n We know, that for an equilateral triangle its Altitude is \\frac{\\sqrt{3}}{2} \\times(\\text { side })<\/span>.<\/span><\/p>\n Hence, the Altitude of the given triangle =\\frac{\\sqrt{3}}{2} \\times 12 \\text { units }=6 \\sqrt{3} \\text { units. }<\/span><\/span><\/p>\n \u2234 The altitude of the given triangle is 6 \\sqrt{3}<\/span> units.<\/span><\/span><\/p>\n The formula for the area of a triangle =\\frac{1}{2}\\text{ base }\\times \\text { height }<\/span><\/span><\/p>\n Base = side length = 12 units and height = altitude = 6 \\sqrt{3} \\text { units }<\/span><\/span><\/p>\n \\therefore \\text { Area }=\\frac{1}{2} \\times 12 \\times 6 \\sqrt{3}=36 \\sqrt{3} \\text { unit }^{2} \\text {. }<\/span><\/span><\/p>\n Hence, the area of the triangle is 36 \\sqrt{3} \\text { unit }^{2}<\/span>.<\/span><\/p>\n <\/span><\/p>\n Example 2<\/strong>:<\/strong> The area of a triangle is 30 sq units. Determine its altitude if the base is 12 units.<\/span><\/p>\n Solution:<\/strong><\/p>\n We know the formula for the area of a triangle, i.e.,<\/span><\/p>\n \\text { Area }=\\frac{1}{2} \\times \\text { Base } \\times \\text { Height }<\/span><\/span><\/p>\n \\Rightarrow \\text { Height }=\\frac{2 \\times \\text { Area }}{\\text { Base }}=\\frac{2 \\times 30}{12}=5 \\text { units }<\/span><\/span><\/p>\n As height and altitude are the same. Hence, the altitude of the given triangle is 5 units in length.<\/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=”4.13.1″ _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\/triangle-one-300x161.png\")
<\/span><\/p>\nProperties of Altitude of a triangle<\/b><\/h2>\n
\n
Altitudes of different types of triangle<\/b><\/h2>\n
Equilateral Triangle<\/b><\/h3>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/triangle-2-300x216.png\")
<\/span><\/p>\n
<\/span><\/p>\n\\sin 60^{\\circ}=\\frac{\\text { perpendicular\/height }}{\\text { hypotenuse }}=\\frac{X O}{X Y}=\\frac{H}{A}<\/span>\nIsosceles Triangle<\/b><\/b><\/h3>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/triangle-3-300x223.png\")
<\/span><\/p>\nRight Angled Triangle<\/b><\/h3>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/triangle-4-300x189.png\")
<\/b><\/p>\n<\/h3>\n
Obtuse Triangle<\/b><\/h3>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/triangle-5-300x135.png\")
<\/span><\/b><\/p>\n<\/b><\/h2>\n
Examples<\/b><\/h2>\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>\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