Answer:
x = √52
Explanation:
Triangle PLM and JLK are connected together at L, which means that they share the same vertex L and therefore, the angle at L is the same for both triangles.
Since PLM and JLK are both triangles, we know that the sum of the angles in each triangle is always 180 degrees.
Let's call the angle at L as a. So we have:
a + (180-a) = 180
We know that PL = 6 and PM = 12, so we can use the Law of Cosines to find the value of the angle a:
c^2 = a^2 + b^2 - 2ab*cos(C)
where c is the hypotenuse, a and b are the other two sides and C is the angle opposite to c.
c = PL = 6
a = PM = 12
b = JL = 4
c^2 = 12^2 + 4^2 - 2124cos(a)
36 = 144 - 48cos(a)
cos(a) = (144 - 36)/48 = (108/48) = 9/4
a = arccos(9/4)
Now we know the measure of angle a and we can use it in the equation of the sum of angles in a triangle.
a + (180-a) = 180
We also know that JL = 4 and JK = x and we can use the Law of Cosines again to find the value of x.
x^2 = 4^2 + 6^2 - 246cos(a)
x^2 = 16 + 36 - 24(9/4)
x^2 = 52
x = √52