We have two right angle triangles, Δs ABD and ADC.
Let's file that away, it will come in handy.
Let's firstly focus on ΔADC, as we can extrapolate more information of it.
Now, since one angle is 90° and the other is 45°, then by definition, the third angle must be the remaining angle adding up to 180°
Let the angle be some variable, u.
90 + 45 + u = 180 (angle sum of triangle)
135 + u = 180
u = 45
Since two angles are equal, then this means that the opposite sides are also equal. Thus, z and y are equal.
Now, we know that ΔADC is a right angle, so we can write an expression for the unknown sides using Pythagoras' Theorem.
z² + y² = 28²
Since z = y, we can eliminate one of the variables.
y² + y² = 28²
2y² = 28²
2y² = 784
y² = 392
y = √392, since y > 0
That means z is also √392 = 14√2
Now, we have a value for z. That seems useful for the other triangle, so let's switch over to ΔABD.
Since we have a right angle triangle, we know that we can use trigonometric identities.
We have a value for z, and we have an angle, so we can use the sine function to find w, or we can find x using tangent function.
Finding w:
sin(30) = z/w
sin(30) = 1/2
So, 1/2 = z/w
w/2 = z
w = 2z
w = 28√2
Now that we have w, we can find x.
Finding x:
tan(30) = z/x
tan(30) = 1/√3
Thus, 1/√3 = z/x
x/√3 = z
x = z/(1/√3)
x = √3 · z
x = √3 · 14√2 = 14√6
Thus, w = 28√2, x = 14√6, y = 14√2, z = 14√2
