Question

Complete the proof of the Law of Sines/Cosines for Triangle ABC with altitude segment AD labeled x.

A) Angles ADB and CDA are altitudes; x = b ⋅ sinB; b ⋅ sinB = c ⋅ sinC
B) Angles ADB and CDA are right angles; x = b ⋅ sinB; b ⋅ sinB = c ⋅ sinB
C) Angles ADB and CDA are altitudes; x = b ⋅ sinB; c ⋅ sinB = b ⋅ sinC
D) Angles ADB and CDA are right angles; x = b ⋅ sinB; c ⋅ sinB = b ⋅ sinC

Answer 1

The Law of Sines states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively. Using this law and the given information in the question, we can see that Angle ADB is an altitude, which means that AD is the height of triangle ABC. The length of AD is labeled x. Using the Law of Sines, we can write the following equations: x/sinB = c/sinC and x/sinB = b/sinA. Simplifying these equations, we get: x = b*sinB and c*sinB = b*sinC.

Step-by-step explanation:

The Law of Sines states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively:

a/sinA = b/sinB = c/sinC

Using this law and the given information in the question, we can see that:

Angle ADB is an altitude, which means that AD is the height of triangle ABC. The length of AD is labeled x.

Using the Law of Sines, we can write the following equations:

x/sinB = c/sinC

and

x/sinB = b/sinA

Simplifying these equations, we get:

x = b*sinB

and

c*sinB = b*sinC

Therefore, option B is the correct answer: Angles ADB and CDA are right angles; x = b*sinB; b*sinB = c*sinC.