Mathematics: Points m and n are the midpoints of sides ac and bc of △abc. prove that the ratio of the area of △mnc to the area of abnm is 1: 3.

please can you keep being amazingStep-by-step explanation:...

Points m and n are the midpoints of sides ac and bc

Ответ:

girlwonder326

10.03.2022

Mathematics

$\dfrac{1}{3}$

Step-by-step explanation:

If points M and N are the midpoints of sides AC and BC of △ABC, then segment MN is a midline segment of the triangle ABC. By the triangle midline theorem, MN║AB and is half of AB.

Since

angle C is common;MN║AB, then angles A and M are congruent and angles B and N are congruent,

then triangles ABC and MNC are similar by AAA theorem with $\dfrac{1}{2}$ factor of similarity. Thus,

$A_{\triangle MNC}=\dfrac{1}{4}A_{\triangle ABC}.$

The area of the quadrilateral ABNM is

$A_{ABNM}=A_{\triangle ABC}-A_{\triangle MNC}=A_{\triangle ABC}-\dfrac{1}{4}A_{\triangle ABC}=\dfrac{3}{4}A_{\triangle ABC}.$

Then

$\dfrac{A_{\triangle MNC}}{A_{ABNM}}=\dfrac{\frac{1}{4}A_{\triangle ABC}}{\frac{3}{4}A_{\triangle ABC}}=\dfrac{1}{3}.$

Points m and n are the midpoints of sides ac and bc of △abc. prove that the ratio of the area of △mn

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Points m and n are the midpoints of sides ac and bc of △abc. prove that the ratio of the area of △mnc to the area of abnm is 1: 3.

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