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AMC 8 USA Math Olympiad

Menalaus Theorem in AMC 8 2019

Learn how to use Menalaus’s Theorem to solve geometry problem from AMC 8 2019. We also provide Knowledge Graph and a video discussion.

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What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Menalaus’s Theorem This problem from American Mathematics contest (AMC 8, 2019) will help us to learn more about Menalaus’s Theorem.ย 

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First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2019/12/AMC-8-2019-Problem-24.png” align=”center” force_fullwidth=”on” _builder_version=”4.0.9″][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]In triangle ๐ด๐ต๐ถ, point ๐ท divides side AC so that ๐ด๐ท โˆถ ๐ท๐ถ = 1 โˆถ 2. Let ๐ธ be the midpoint of BD and ๐น be the point of intersection of line BC and line AE. Given that the area of โˆ†๐ด๐ต๐ถ is 360, what is the area of โˆ†๐ธ๐ต๐น?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0.9″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.0.9″ open=”off”]American Mathematical Contest 2019, AMC 8 Problem 25

[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” open=”on” _builder_version=”4.0.9″]Menalaus’s Theorem:ย ย  Given a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C, then

$$ \displaystyle {\frac {AF}{FB}\times \frac {BD}{DC}\times \frac {CE}{EA}=-1.}$$

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7/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematicsย 

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Watch video

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Connected Program at Cheenta

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Similar Problems

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By Ashani Dasgupta

Ph.D. in Mathematics from University of Wisconsin Milwaukee (USA)
Founder - Faculty at Cheenta

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