Final answer:
The correct option is C). The cosine rule, also known as the law of cosines, is a formula used to find the length of a side or the measure of an angle in a triangle. It can be derived using the point-to-line distance formula. The derivation involves expressing the distances from a point to the sides of the triangle using the cosine rule and substituting them into the point-to-line distance formula. The resulting equation is simplified, expanded, and rearranged to obtain the cosine rule, which states that the square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice their product and the cosine of the included angle.
Step-by-step explanation:
The cosine rule, also known as the law of cosines, is a formula used to find the length of a side or the measure of an angle in a triangle. It can be derived using the point-to-line distance formula. Here are the steps to prove the cosine rule using the point-to-line distance formula:
Consider a triangle ABC with sides a, b, and c, and let P be a point on side c.
Using the point-to-line distance formula, the distance from point P to line AB is given by: d = |(AP x BP)| / |AB|, where AP and BP are the distances from point P to points A and B respectively, and |AB| represents the length of side AB.
Now, let's express AP and BP in terms of a, b, and c. Using the cosine rule, we have AP = sqrt(b^2 + d^2 - 2bd * cos(C)) and BP = sqrt(a^2 + d^2 - 2ad * cos(C)).
Substituting the values of AP and BP into the point-to-line distance formula, we get: d = [sqrt(b^2 + d^2 - 2bd * cos(C))] * [sqrt(a^2 + d^2 - 2ad * cos(C))] / |AB|.
Simplifying the equation, we have: d * |AB| = sqrt((b^2 + d^2 - 2bd * cos(C))(a^2 + d^2 - 2ad * cos(C))).
Squaring both sides of the equation, we get: d^2 * |AB|^2 = (b^2 + d^2 - 2bd * cos(C))(a^2 + d^2 - 2ad * cos(C)).
Expanding and rearranging the equation, we have: d^2 * (a^2 + b^2 + c^2) = a^2 * b^2 - 2ab * cos(C) + b^2 * c^2 - 2bc * cos(A) + a^2 * c^2 - 2ac * cos(B).
Since d = c, we can substitute it in the equation, resulting in c^2 * (a^2 + b^2 + c^2) = a^2 * b^2 - 2ab * cos(C) + b^2 * c^2 - 2bc * cos(A) + a^2 * c^2 - 2ac * cos(B).
Canceling out the common terms, we get: c^2 = a^2 + b^2 - 2ab * cos(C). This is the cosine rule.