Answer:
11.33cm
Explanation:
We can solve this by first solving for angle ∠ADB, then calculate angle ∠CDB, then angle ∠CBD using that, and finally solve for x from there.
We know we can do this because the law of cosines enables us to solve for an angle in a triangle if we know all three sides, and because AC is a straight line, we know that ∠ADB + ∠CDB = 180°. Then, because the angles of a triangle add up to 180 degrees, we can solve for ∠CBD, and using the law of sines, knowing one side (BD = 11 cm) and all three angles in a triangle (BDC), we can solve for any side based on that.
First, we can solve for ∠ADB. The law of cosines states that
c² = a² + b² - 2ab * cos(c). Each side corresponds to the angle opposite of it, and as a result, we can tell that ∠ADB is opposite side AB. Therefore,
AB² = BD² + AD² - 2(BD)(AD)(cos(∠ADB))
13² = 11² + 5² - 2(11)(5)(cos(∠ADB))
169 = 121 + 25 - 110(cos(∠ADB))
169 = 146 - 110(cos(∠ADB))
subtract 146 from both sides to isolate the variable and its coefficient
23 = -110(cos(∠ADB))
divide both sides by -110 to isolate the variable
-23/110 = cos(∠ADB)
arccos(-23/110) = ∠ADB ≈ 102.07°
Then, 180 = ∠CDB + ∠ADB
180 - ∠ADB = ∠CDB
180 - 102.07 = 77.93
angles in a triangle add to 180 degrees
∠CBD + ∠CDB + ∠BCD = 180
∠CDB + 77.93 + 50 = 180
180 - 50 - 77.93 = ∠CBD = 52.07
law of sines:
a/sin(A) = b/sin(B) = c/sin(C), where corresponding angles are opposite of their sides
11/sin(50) = BC/sin(77.93) = x/sin(52.07)
we want to solve for x, so there should only be one unknown
11/sin(50) = x/sin(52.07)
multiply both sides by sin(52.07) to isolate x
11 * sin(52.07)/sin(50) = x = 11.33 cm