Question

select the correct answer from each drop-down menu.
a student begins a proof of the law of cosines. his work is shown.
triangle diagram with labels: y at top, x and z at base, a is foot of perpendicular from y to xz, ax = z cos(x), xz = y, sides xy = z, yz = x
step 1: \\(\cos(x) = \frac{ax}{z}\\)
\\(z\cos(x) = ax\\)
the next step in the proof is to write an expression for the length of side az in terms of drop-down
then the student should write an expression for the length of side ay in terms of drop-down
next he can use the drop-down to relate x, y, z, and drop-down.

Explanation:

Step1: Find AZ length

AZ = XZ - AX = y - zcos(X)

Step2: Find AY length

Use sine for AY: sin(X) = AY/z → AY = zsin(X)

Step3: Choose theorem to relate sides

Right triangle AYZ uses Pythagorean theorem: AY² + AZ² = YZ²

Step4: Identify angle in relation

Law of cosines relates to angle X

Answer:

1. y - zcos(X)
2. zsin(X)
3. Pythagorean theorem
4. angle X