{"id":5836,"date":"2021-12-15T10:08:37","date_gmt":"2021-12-15T10:08:37","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=5836"},"modified":"2022-01-03T07:34:18","modified_gmt":"2022-01-03T07:34:18","slug":"centroid-formula-derivation-solved-examples","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/centroid-formula-derivation-solved-examples\/","title":{"rendered":"Centroid Formula – Derivation -Solved Examples"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.13.1″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n

Centroid Formula – Derivation -Solved Examples<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

Centroid Formula<\/b><\/h2>\n

The centroid of a triangle is the point where all three medians of a triangle intersect each other. It is also known as the geometric centre and always lies inside the triangle.<\/span><\/p>\n

This point divides every median of the triangle in the ratio of 2:1.<\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

The coordinates of all the vertices of \u25b3ABC are given below<\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

G is the centroid of \u25b3ABC<\/span><\/p>\n

Centroid Formula for \\triangle \\mathrm{ABC}=\\left(\\frac{x_{a}+x_{b}+x_{c}}{3},\\frac{y_{a}+y_{b}+y_{c}}{3}\\right)<\/span><\/span><\/p>\n

<\/span><\/p>\n

Derivation<\/b><\/h2>\n

In \u25b3ABC<\/span><\/p>\n

A, B and C are the vertices having x coordinates and y coordinates as shown below<\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

P – Midpoint of BC<\/span><\/p>\n

Q – Midpoint of CA<\/span><\/p>\n

R – Midpoint of AB\u00a0<\/span><\/p>\n

Using section formula, the coordinates of P, Q and R are calculated as shown in the table below:<\/span><\/span><\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

The point G divides AP in the ratio of 2:1.<\/span><\/p>\n

Using section formula,<\/span><\/p>\n

x coordinate of G =\\frac{2\\left(\\frac{x_{b}+x_{c}}{2}\\right)+1\\left(x_{a}\\right)}{2+1}=\\frac{x_{a}+x_{b}+x_{c}}{3}<\/span><\/span><\/p>\n

y coordinate of G =\\frac{2\\left(\\frac{y_{b}+y_{c}}{2}\\right)+1\\left(y_{a}\\right)}{2+1}=\\frac{y_{a}+y_{b}+y_{c}}{3}<\/span><\/span><\/p>\n

Hence it is proved that the coordinates of centroid G of a triangle are:<\/span><\/p>\n

\\left(\\frac{x_{a}+x_{b}+x_{c}}{3},\\frac{y_{a}+y_{b}+y_{c}}{3}\\right)<\/span><\/span><\/p>\n

<\/span><\/p>\n

Solved Examples<\/b><\/h2>\n

<\/b><\/p>\n

1. The vertices of a triangle are (0,3), (5,6) and (1,2). Find its centroid.<\/strong><\/p>\n

Solution:<\/strong><\/p>\n

\"\"<\/span><\/p>\n

Let the centroid of \u25b3ABC be G<\/span><\/p>\n

Using the centroid formula,<\/span><\/p>\n

x coordinate of G =\\frac{x_{a}+x_{b}+x_{c}}{3}=\\frac{0+5+1}{3}=2<\/span><\/span><\/p>\n

y coordinate of G =\\frac{y_{a}+y_{b}+y_{c}}{3}=\\frac{3+6+2}{3}=\\frac{11}{3}<\/span><\/span><\/p>\n

Therefore, the coordinates of the centroid are \\left(2, \\frac{11}{3}\\right)<\/span>.<\/span><\/p>\n

<\/span><\/p>\n

2. The centroid of \u25b3ABC is (0,3). <\/strong>Find the coordinates of point A if the other two points are\u00a0 (-2,4) and (2,2).<\/strong><\/p>\n

Solution:<\/strong><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

Using centroid formula\u00a0<\/span><\/p>\n

x coordinate of G =\\frac{x_{a}+x_{b}+x_{c}}{3}=\\frac{x-2+2}{3}<\/span><\/span><\/p>\n

\n\\Rightarrow \\frac{x-2+2}{3}=0 <\/span>
\\Rightarrow x=0\n<\/span><\/p>\n

y coordinate of \\mathrm{G}=\\frac{y_{a}+y_{b}+y_{c}}{3}=\\frac{y+4+2}{3}<\/span><\/span><\/p>\n\\Rightarrow \\quad \\frac{y+2+2}{3}=3<\/span>\n

\\Rightarrow y+4=9<\/span><\/span><\/p>\n

\\therefore y=9-4=5<\/span><\/span><\/p>\n

Hence the coordinate of point A is (0,5).<\/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
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Explore Other Topics<\/h1>\n

[\/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

\n
Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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Orthocentre<\/a><\/div>\n
Circumcentre<\/a><\/div>\n
Concurrent Lines<\/a><\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1. Define the centroid of a triangle?
    <\/strong><\/h3>\n

    Ans: <\/strong><\/span>The point where all three medians of a triangle meet each other is termed the centroid of the triangle.<\/span><\/strong><\/strong><\/span><\/span><\/p>\n

    Q2. What is the centroid formula?
    <\/strong><\/h3>\n

    Ans: <\/strong>It is a formula for the coordinates of the centroid of a triangle.<\/span><\/p>\n

    The centroid formula of a triangle is \\left(\\frac{x_{a}+x_{b}+x_{c}}{3},\\frac{y_{a}+y_{b}+y_{c}}{3}\\right)<\/span><\/span>, <\/span>where \\left(x_{a}, y_{a}\\right),\\left(x_{b}, y_{b}\\right) \\text { and }\\left(x_{c}, y_{c}\\right)<\/span> <\/span>are the coordinates of the vertices of the triangle.<\/span><\/span><\/p>\n

    Q3. Which formula is used to derive the centroid formula?
    <\/strong><\/h3>\n

    Ans: <\/strong>Section formula is used to derive the centroid formula.<\/span><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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