Question 1070529
Here is triangle ABC with midsegments LM and MN {{{drawing(360,300,-5,7,-1,9,
green(rectangle(0,0,3,4)),
red(triangle(0,0,6,0,0,8)),
line(0,0,3,4),
locate(3,4.5,M),locate(6,0,B),
locate(-0.2,0,A),locate(-0.1,8.5,C),
locate(-0.3,4.2,L),locate(2.9,0,N)
)}}}
You know that those midsegments are parallel to
and half the length of the side they do not touch.
That makes ALMN a rectangle.
Line AM is the diagonal of rectangle ALMN.
Triangle ABC is a {{{red(3)}}} times scaled-up version of the most popular 3-4-5 right triangle,
with side lengths in cm of
{{{AB=3*red(2)=6}}} ,
{{{AC=4*red(2)=8}}} , and
{{{BC=5*red(2)=10}}} .
It has a smaller acute angle ACB (opposite shorter side AB),
a larger acute angle ABC, and a right angle BAC (opposite longest side BC).
The mid segments, and AM split triangle ABC into four smaller triangles.
They are all 3-4-5 right triangles with sides measuring
{{{LM=AN=NB=3cm}}} ,
{{{AL=LC=CM=4cm}}} ,
{{{BM=MC=highlight(AM=5cm)}}} ,
congruent smaller acute angles (so that {{{CAM=ACM}}} for example),
and congruent larger acute angles, as well.
Since they tell us that "MD and intersects AC at point F",
I know to what side of segment AM square AMDE is locate,
and I can draw the square, like this
{{{drawing(360,300,-5,7,-1,9,
green(rectangle(0,0,3,4)),
red(triangle(0,0,6,0,0,8)),
line(0,0,3,4),line(-4,3,-1,7),
line(0,0,-4,3),line(3,4,-1,7),
locate(3,4.5,M),locate(6,0,B),
locate(-0.2,0,A),locate(-0.1,8.5,C),
locate(-4.2,3,E),locate(-1.1,7.5,D),
locate(-0.3,6.3,F)
)}}} {{{area(AFDE)=area(AMDE)-area(MFA)}}}
{{{area(AMDE)=AM^2=(5cm)^2=25cm^2}}}
and triangle MFA is a right triangle,
similar to LMA and all the other 3-4-5 triangles we looked at before,
because it shares the same smaller acute angle CAM with triangle LMA.
So, for MFA, the shorter leg to longer leg ratio is the same
{{{MF/AM=3/4}}} --> {{{MF/"3 cm"=3/4}}} --> {{{MF=(3/4)(3cm)}}} --> {{{MF=9/4}}}{{{cm)}}}
{{{area(MFA)=(1/2)*AM*MF}}}
{{{area(MFA)=(1/2)*3*(9/4)}}}{{{cm^2}}}
{{{area(MFA)=27/8}}{{{cm^2}}}
{{{area(AFDE)=area(AMDE)-area(MFA)}}}
{{{area(AMDE)=25cm^2-3.375cm^2=highlight(21.625cm^2)}}}