In triangle XYZ, angle X measures 60 degrees, side XY is 5 units long, and side YZ is 7 units long. Find the length of side XZ.

Identify Given Information:

→ Angle X measures 60 degrees.
→ Side XY is 5 units long.
→ Side YZ is 7 units long.

Apply the Law of Cosines:
The Law of Cosines states: \$"c"^2 = "a"^2 + "b"^2 - 2"ab"."cos"("C")\$ where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.

Substitute Given Values:
Let’s denote the length of side XZ as c, the length of side XY as a, and the length of side YZ as b.
Substituting the given values, we have:

\$"c"^2 = 5^2 + 7^2 - 2(5)(7) . "cos"(60)\$

Calculate Cosine of 60 Degrees:

The cosine of 60 degrees is \$1/2\$

Simplify the Equation:

\$"c"^2 = 25 + 49 - 70 . 1/2\$
\$"c"^2 = 25 + 49 - 35\$
\$"c"^2 = 74 - 35\$
\$"c"^2 = 39\$

Take the Square Root of Both Sides:

\$"c" = \sqrt(39)\$

So, the length of side XZ is \$\sqrt(39)\$.

This corresponds to the length of side XZ calculated using the Law of Cosines, as demonstrated in the solution steps

\$\sqrt39\$

This value differs from the calculated value of \$\sqrt(39)\$. Therefore, option B is not correct

\$\sqrt35\$

This option is not the correct answer. It’s different from the calculated length \$\sqrt(39)\$

\$\sqrt32\$

This option represents the square root of 29, which is different from the length we found for side XZ

\$\sqrt29\$

Option

A

Can you summarize what you’ve understood in the above steps?