Can you find the Length X in this Triangle? | Step-by-Step Explanation

#OlympiadMathematics #OlympiadPreparation #CollegeEntranceExam

[mohanramachandran]

Very very excellent question &
Easy solutions. Thank you.

[jayquirk]

Nice question. As an alternative solution, I did the following. Detach and rotate triangle MNB and attach side BM to CM, which are equal. You end up with a trapezoid where the left side is 2x. This trapezoid has bottom angles of alpha and each top angle of 180-alpha. This creates an isosceles trapezoid where the right and left sides are the same. The right side length is AC=24. The left side is 2x, therefore x=12.

[daviebowie]

From M we draw a line parallel to AB that intersects with AC at P. As a result, APMN is a trapezium with two equal base angles, so it is Isosceles, then MN=AP. Also, applying Thales theorem to the triangle ABC with MP//AB, CM/BC=CP/AC=1/2, which means P is the middle point of AC, therefore MN=AP=12.

It's just another way into the chase.

[chrisrowland]

Redraw triangle ABC with angle alpha=60. Triangle ABC is now equilateral with side lengths 24. MN lies on line CB with its' endpoints on the midpoint and the vertex of CB. So MN = 24/2 = 12.

[Waldlaeufer]

I really thought about this task for a long time. From the beginning I had the invisible isosceles triangle in mind, which protrudes from the top of the triangle shown. But I didn't get any further.

When you drew in the line PM, the solution was immediately clear to me:

- Because of the ray theorem, PM = 1/2 AC
- Because of the angles in the isosceles triangle, x = PM
- x is therefore always half of 24, i.e. 12.

A wonderful task. Unfortunately, I didn't find the solution myself. I needed the idea with the straight line PM. Thank you. Now let's watch the rest of the video...

[theoyanto]

You've taught me so much geometry
And all of it completely free
Explaining every little part
Not lines or angles but perfect art
The symbols fall before your hand
You make it easy to understand
Blowing away the mystery
Making it clear for all to see
It seems to me it gives you pleasure
To me and others it's the greatest treasure

[neilmorrone]

This exercise nicely introduces and explains several Geometric Principles/Theorems of Triangles (involving its angles and sides) that are important and relevant to solving many Geometry questions! Thank you for using compound concepts! Well done.

[HappyFamilyOnline]

Nicely explained👍
Thank you so much for sharing your knowledge🎉🎉

[mohanramachandran]

I Recommended to so many School students, They are most impressed with your solutions .
Thank you.

[HeyPainting]

Really great sharing! Thanks for your nice video!!~~🙂

[rahulpaul]

Wow, you are using amazing method to solved this midpoint theorem make it easy approach to find the lenght

[guyhoghton]

Move point B towards point A along the line BA, keeping M at the mid point of BC and ∠MNA equal to ∠CAN as B moves. When B reaches A the point N will be coincident there too, BC will coincide with AC and therefore point M will lie at the mid point of AC which remains at 24 units. Therefore since the properties of △ABC were preserved in this operation,  x = MN = ½AC = 12.

[pk]

Before I watched this video, I solved it another way . Draw a line from C to P that is parallel to MP . We then have triangle ACP is isosceles ; so, length CP = 24 . We then have triangles BCP and BMN similar . Call length BM =a and MC =a . We then have length BC = a+a=2a . We then have from similar triangles ---> MN/BM=CP/BC ---> x/a=24/2a ---> x =(24/2a)(a)= 12 .

[AnonimityAssured]

A fine explanation of a rather easy problem. I thought of it in a rather different way. I imagined the small triangle rotated anticlockwise about M, until A and B corresponded, forming a symmetrical trapezium with base AN. Clearly, _x_ would have to be half the length of the right-hand side of that trapezium, which would also be half the length of the already-given left-hand side.

[jhill]

I was having a problem starting this. But when you made the parallel line and mentioned congruent angles, I went "aaagh!" Now so simple. Thanks!

[santiagoarosam]

Draw through M a line parallel to AN, which intersects AC at the point P → If ∠CAN=∠ANM=α → AP=MN=X → AC=AP+PC=X+(24-X) → With center in M, we rotate MN through an angle of 180º → M remains in place and N rotated becomes N´→ NM=MN´=X and both segments are collinear → PC=MN´→ (24-X)=PC=MN´= MN=X → (24-X)=X → 24=2X → X=24/2=12
Thanks and greetings to all

[drpkmath]

Hey my friend! You have a very nice playlist of geometry questions! This one is also very nice! Keep rocking my friend haha

[giuseppemalaguti]

Molto semplice, basta considerare il teorema dei seni per i 2 triangoli NMB e ABC.. x/sinB=MB/sin(a)... memdro a membro diventa x/24=1/2 cioè x=12

[ohao]

From C, draw d parallel to both AB and cut MN at k => to see that NKCA is an isosceles trapezoid AND MK = NK => NK /2=CA/2 = NM = 12
is it right ?

[Q_from_Star_Trek]

Very nice, but I have a simpler solution:
Draw MM' || AB...since M midpoint => M' midpoint, so since AC=24 => AM' =12. Now AM'MN is an trapezoid (MM' || AN) with the base angles alpha (M'AN=MNA=alpha) which makes it an isosceles trapezoid, so AM' = 12 = MN = x.