Answer:
x = 4
y = 6
Step-by-step explanation:
In a triangle, the number of arcs at each vertex indicates angle congruency. So, the measures of the angles marked with one arc are equal to each other, and the measures of the angles marked with two arcs are equal to each other. This means that the line segments drawn from one vertex to the opposite side in each of the given triangles are angle bisectors.
To find the values of x and y in the given triangles, we can use the Angle Bisector Theorem.
The Angle Bisector Theorem states that in a triangle, an angle bisector divides the opposite side into segments proportional to the lengths of the other two sides:
[tex]\boxed{\sf \dfrac{Segment\;1}{Segment\;2}=\dfrac{Side \;1}{Side\;2}}[/tex]
[tex]\dotfill[/tex]
Triangle 1
In the case of the first triangle:
- Segment 1 = x
- Segment 2 = 6 - x
- Side 1 = 8
- Side 2 = 4
Substitute these values into the equation and solve for x:
[tex]\dfrac{x}{6-x}=\dfrac{8}{4}\\\\\\\dfrac{x}{6-x}=2\\\\\\x=2(6-x)\\\\\\x=12-2x\\\\\\3x=12\\\\\\x=4[/tex]
Therefore, the value of x:
[tex]\Large\boxed{\boxed{x=4}}[/tex]
[tex]\dotfill[/tex]
Triangle 2
In the case of the second triangle:
- Segment 1 = 4
- Segment 2 = 6
- Side 1 = y
- Side 2 = 9
Substitute these values into the equation and solve for y:
[tex]\dfrac{4}{6}=\dfrac{y}{9}\\\\\\9 \cdot 4 = 6 \cdot y\\\\\\36=6y\\\\\\y=6[/tex]
Therefore, the value of y is:
[tex]\Large\boxed{\boxed{y=6}}[/tex]
