#triangle problem

50 messages · Page 1 of 1 (latest)

turbid night
#

I want some hints

IMG_20240811_213515_687.jpg
hollow cragBOT
#
  1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  2. Wait patiently for a helper to come along.
  3. Once someone helps you, say thank you and close the thread with:
    +close
  4. Feel free to nominate the person for helper of the week in #helper-nominations
  5. Do not ping the mods, unless someone is breaking the rules.
  6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:
storm river
#

well i assume one of the major steps would be noting that ABC and XBY are similar

#

let me try to give it a shot

#

ok yeah i solved it

#

so similar triangles and theorem about tangential quadrilateral's sides solve this thing

#

ping me for further hints

turbid night
#

And what theorem. I remember some area problems only

storm river
#

well if you have this quarilateral

images.png
#

with an inscribed circle

turbid night
#

Yes?

storm river
#

then you have an equality that ties sides of this shape

#

that is a + c = b + d

turbid night
#

Quadrilaterla made of tangent

turbid night
storm river
#

so i proved that these triangles are similar, applied this theorem on some quadrilateral, used similarity coefficient somehow

turbid night
storm river
#

yeah so

turbid night
#

Similar triangle problem

storm river
#

XY + AC = CX + AY, do you see that?

turbid night
#

Yes

turbid night
storm river
#

right

#

also do you see similar triangles here?

turbid night
#

Of course

storm river
#

these are?

turbid night
#

xb/yb = 4/3?

storm river
#

well uhh yeah but thats not relevant

#

brb

turbid night
#

Please help

#

Similar triangle

storm river
#

well what this means is that there exists a number k such that:
XY = k * AC
BX = k * BC
BY = k * AB

#

do you see why?

turbid night
storm river
#

and rewrite it so that its only in terms of k and lengths of sides we already know

#

try to do that yourself please

turbid night
#

XY/2=BX/4=By/2=k

#

XY=CX+Ay-2

storm river
#

uhhh

#

well not helpful

#

ok so look

#

XY = k * AC = 2k
AC = 2
CX = BC - BX = BC - BC * k = 4 - 4k
AY = AB - BY = AB - AB * k = 3 - 3k

#

do you see why that is true?

turbid night
#

Yes

#

2k=4-4k+3-3k-2

k=5/9

#

xy=10/9

turbid night
#

+close