Apply the Pythagorean Theorem to each right triangle and relate the equations. Also, find the corresponding trigonometric ratio.
See solution.
Practice makes perfect
We will prove the Law of Cosines by filling in the blanks for the given column proof. Let's first recall what this law says.
To prove this law, we begin by drawing the altitude from A.
Notice that the altitude separates △ ABC into two right triangles. These are the first two steps written in the given table. Now, we will continue by filling in the blanks.
Filling in the Blanks
We start applying the Pythagorean Theorem to the left-hand side triangle.
3)& c^2 = ( a- x)^2+ h^2
3)& a. Pythagorean Theorem
Next, we expand the perfect square.
4)& c^2 = a^2-2 a x+ x^2+ h^2
4)& b. Expanding the square
Once again, we apply the Pythagorean Theorem but this time to the right-hand triangle.
5)& x^2+ h^2 = b^2
5)& c. Pythagorean Theorem
Let's substitute the latter expression into the one obtained in step 4.
6)& c^2 = a^2-2 a x+ b^2
6)& d. substitution
Now, we find the cosine of ∠ C.
7)& cos C = x/b
7)& e. definition of cosine
We solve the latter equation for x.
8)& bcos C = x
8)& f. solving for x
Let's substitute this expression of x into the equation obtained in step 6.
9)& c^2 = a^2-2 a( bcos C) + b^2
9)& g. substitution
Finally, by using the commutative property of addition, we rearrange the latter equation to obtain the required result.
10)& c^2 = a^2+ b^2 -2 a bcos C
10)& h. commutative property
Complete Proof
In the table below, we write the entire proof.
Statements
Reasons
1.
h is an altitude of △ ABC
1.
Given
2.
Altitude h separates △ ABC into two right triangles
