Let's begin by writing the given information and what we want to prove.
Given: & BDis an altitude ofâ–ł ABC.
Prove: & a^2 = b^2+c^2 - 2bccos A
We will use the following diagram given with the proof.
Step 1
In every proof we do, the first step is to write the given information.
1)& BDis an altitude ofâ–ł ABC
1)& Given
Step 2
By definition of altitude, we have that both â–ł ADB and â–ł CDB are right triangles.
2)& â–ł ADB and â–ł CDB are right triangles
2)& Definition of altitude
Step 3
By applying the Pythagorean Theorem to â–ł CDB we get that a^2=(b-x)^2+h^2.
3)& a^2=(b-x)^2+h^2
3)& Pythagorean Theorem
Step 4
In this step, we expand the binomial written in the previous step.
Then, we can write the missing statement in step 6.
6)& a^2 = b^2 + c^2 - 2bx
6)& Substitution Property of Equality
Step 7
Since â–ł ADB is a right triangle, we can use the cosine ratio to get the seventh statement.
7)& cos A = x/c
7)& Definition of Cosine ratio
Step 8
In this step, we multiply both sides of the equation obtained in the previous step by c. We do this in order to isolate x.
8)& x = ccos A
8)& Multiplication Property of Equality
Step 9
Finally, we substitute x=ccos A (obtained in the previous step), into the equation written in step 6. That way, we obtain the Law of Cosines.
9)& a^2 = b^2+c^2 - 2bccos A
9)& Substitution Property of Equality
Complete Proof
In the following two-column table we summarize all the steps written before.
