Question 1202263
<font color=black size=3>
Answer: 36/13
This fraction approximates to 2.7692


============================================================================================


Explanation:


There are at least two ways to solve this problem.


Method 1


This is one way to draw things out
<img src = "https://i.imgur.com/BZUWrPr.png">
I used <a href="https://www.geogebra.org/">GeoGebra</a> to make the diagram.
c is a real number such that 0 < c < 10.


As mentioned by the other tutor @greenestamps, we can use the <a href="https://en.wikipedia.org/wiki/Angle_bisector_theorem">angle bisector theorem</a> to set up this proportion
PQ/QX = PR/XR


So,
PQ/QX = PR/XR
9/c = 17/(10-c)
9(10-c) = 17c
90-9c = 17c
90 = 17c+9c
90 = 26c
c = 90/26
c = 45/13


Therefore,
QX = c = 45/13
XR = 10-c = 85/13


Also mentioned by @greenestamps is <a href="https://en.wikipedia.org/wiki/Heron%27s_formula">Herons Formula</a> to get the area of triangle PQR to be exactly 36 square units.


Notice that triangles PQX and PXR have the same height. 
QR is horizontally flat on the ground.
Let h be the unknown height of each smaller triangular piece.
I'll then let A and B represent the areas of triangles PQX and PXR respectively.


A = area of triangle PQX 
A = (1/2)*base*height
A = (1/2)*QX*h
A = (1/2)*(45/13)*h
A = (45/26)*h


B = area of triangle PXR
B = (1/2)*base*height
B = (1/2)*XR*h
B = (1/2)*(85/13)*h
B = (85/26)*h


Areas A and B must add up to 36 which is the area of triangle PQR.


A+B = area of triangle PQR
(45/26)*h + (85/26)*h = 36
(45/26+85/26)*h = 36
((45+85)/26)*h = 36
(130/26)*h = 36
5*h = 36
h = 36/5



So,
area of triangle PXR = (1/2)*base*height
area of triangle PXR = (1/2)*XR*h
area of triangle PXR = (1/2)*(85/13)*(36/5)
area of triangle PXR = 306/13


Rotate triangle PXR so that PR is horizontal and acts as the base. 
<img src = "https://i.imgur.com/rGDWpwV.png">
Segment XY is the height of triangle PXR since it is perpendicular to the base PR.


area of triangle PXR = (1/2)*base*height
306/13 = (1/2)*PR*XY
306/13 = (1/2)*17*XY
XY = (306/13)*2*(1/17)
XY = 36/13
XY = 2.7692 approximately


==================================================================================


Method 2


Here's another way to draw things out. 
<img src = "https://i.imgur.com/xSJAOQa.png">
The sides of the triangle are
PQ = 9
QR = 10
PR = 17


I've split side PR into two pieces
PY = m and YR = 17-m
where 0 < m < 17.


Focus on triangle PQR.
Use the law of cosines to determine angle R.
c^2 = a^2 + b^2 - 2*a*b*cos(C)
r^2 = p^2 + q^2 - 2*p*q*cos(R)
(PQ)^2 = (QR)^2 + (PR)^2 - 2*(QR)*(PR)*cos(R)
(9)^2 = (10)^2 + (17)^2 - 2*(10)*(17)*cos(R)
81 = 389 - 340*cos(R)
81-389 = -340*cos(R)
-308 = -340*cos(R)
cos(R) = -308/(-340)
cos(R) = 77/85
R = arccos(77/85)
R = 25.057615418303
R = 25.0576
This value is approximate.
Your calculator needs to be in degree mode.


Follow similar steps to find angle QPR = 28.0725 degrees approximately.


Angle bisector PX will split angle QPR into two equal pieces angle QPX and angle XPY
angle QPX = angle XPY = 28.0725/2 = 14.03625 


Let's focus our attention on triangle XPY.
This is a right triangle, so we can use one of the <a href="https://www.mathsisfun.com/algebra/sohcahtoa.html">SOH CAH TOA trig ratios</a>.
I'll use tangent so we can connect PY and XY.


tan(angle) = opposite/adjacent
tan(P) = XY/PY
PY*tan(P) = XY
XY = PY*tan(P)
XY = m*tan(14.03625)
XY = m*0.25000012
XY = 0.25000012m


Now focus on triangle XYR
tan(angle) = opposite/adjacent
tan(R) = XY/YR
YR*tan(R) = XY
XY = YR*tan(R)
XY = (17-m)*tan(R)
XY = (17-m)*tan(25.0576)
XY = (17-m)*0.46753214


The conclusions of the previous two paragraphs found that
XY = 0.25000012m
and
XY = (17-m)*0.46753214


Set them equal to one another and solve for m
I'll skip the steps, but the solution to 0.25000012m = (17-m)*0.46753214 is roughly m = 11.0769


This means:
PY = m = 11.0769
YR = 17-m = 17-11.0769 = 5.9231
and furthermore
XY = 0.25000012*m
XY = 0.25000012*11.0769
XY = 2.769226329228
XY = 2.7692


Or you could compute it like this
XY = (17-m)*0.46753214
XY = (17-11.0769)*0.46753214
XY = 2.769239618434
XY = 2.7692


The drawback of this second method is that we cannot arrive at a nice fraction we did with method 1. 
But it is often good practice to be able to solve a problem multiple ways. 
</font>