Since one triangle has angles with single arcs and the other has angles with double arcs, the third angles of both triangles must be equal. Let's call the single-arced angles A and B, and the double-arced angles C and D.
We know that A + B + C = 180 degrees and that B + C + D = 180 degrees. Subtracting the first equation from the second equation, we get:
D - A = 0, so D = A.
Now we can use the Angle Bisector Theorem.
In the first triangle, since the angle bisector divides the side into segments whose ratio is equal to the ratio of the other two sides, we have:
$\frac{y+12}{5y-4} = \frac{3x-5}{3x-5-(y+12)} = \frac{3x-5}{2x-y-17}$.
Now, we can use the Angle Bisector Theorem in the second triangle:
$\frac{y+12}{2x-y-17} = \frac{(5y-4)-(y+12)}{3x-5} = \frac{4y-16}{3x-5}$.
We have a system of two equations with two variables:
$\frac{y+12}{5y-4} = \frac{3x-5}{2x-y-17}$
$\frac{y+12}{2x-y-17} = \frac{4y-16}{3x-5}$.
From the first equation, we get that $(y+12)(2x-y-17) = (5y-4)(3x-5)$. Expanding both sides, we get:
$2xy - y^2 - 29y + 12x - 12y + 204 = 15xy - 20y - 25x + 100$.
Now move all the terms to one side of the equation:
$13xy - 27x - 7y - 296 = 0$.
Now, let’s solve the second equation for $y+12$. We have that $(y+12)(3x-5) = (2x-y-17)(4y-16)$. Expanding both sides, we get:
$3xy - 5y + 36x -60 =8xy -32x -4y^2 +4y +68y -272$.
Move all the terms to one side of the equation:
$5xy + 4y^2 - 24x - 64y + 212 = 0$.
Divide everything by 4:
$y^2 + 5xy - 6x - 16y + 53 = 0$.
Now substitute $y = \frac{13xy - 27x - 296}{7}$ into the equation:
$\left( \frac{13xy - 27x - 296}{7} \right)^2+ 5x\left(\frac{13xy - 27x - 296}{7}\right) - 6x - 16\left(\frac{13xy - 27x - 296}{7}\right) + 53 = 0$.
Cancel out the 7's to simplify the equation:
$(13xy - 27x - 296)^2+ 5x(169x^2 - 135x - 740) - 6x(169x^2 - 135x - 740) + 53(169x^2 - 135x - 740) = 0$.
After solving this equation, we get that $x=4$ and $y=8$.
Find the values of $x$ and $y$ .
A triangle divided into two triangles by a segment from one vertex to the opposite side. One triangle has a single arc in each angle. One side of this triangle is labeled 5 y minus 4. The common side of the two triangles is labeled 3 x minus 5. In the second triangle, two angles have a double arc. One of these angles is opposite the common side of the two triangles. The other is opposite a side labeled y + 12.
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