In triangle ABC with AM as an angle bisector, the values are AB (x) = 12.5, MC (y) ≈ 21.65, and AC (z) ≈ 21.65, calculated using the properties of 30-60-90 and 45-45-90 triangles.
Given that AM is the angle bisector of <ABC, we can deduce that triangle ABC is a right angle triangle with angle ABC = 90 degrees.
Since AM is also an angle bisector, it separates angle CAM into two equal angles of 45 degrees each.
Thus, triangle AMC is an isosceles right triangle, which implies AC = MC = z = y.
Since angle ABM is 60 degrees and angle AMB is 90 degrees, then triangle ABM is a 30-60-90 triangle.
In such a triangle, the sides are in the ratio 1:√3:2, and considering that BM = 25, the side AB opposite the 30-degree angle will be half of BM (hypotenuse).
Therefore, AB = x = 1/2 * 25 = 12.5.
Using the properties of the 45-45-90 triangle for triangle AMC, we know that AC = MC, hence the value of y and z are equal.
Since angle BMC is 90 degrees and we've found AB (x) to be 12.5 and BM to be 25, we apply the Pythagorean theorem to determine MC (y).
So, MC = y = √(BM2 - AB2) = √(252 - 12.52) = √(625 - 156.25) = √468.75 ≈ 21.65
Therefore, the values are x = 12.5, y ≈ 21.65, z ≈ 21.65.
The probable question may be:
In triangle ABC, AM is angle bisector to line BC.
angle ABM=60 degree, angle AMB=90 degree, angle BAM=30 degree.
angle ABC=90 degree, angle MAC=45 degree, angle ACM=45 degree
AB=x, AC= z, MC=y, BM=25, Find the value of x, y and z