Question

follow these steps to derive the law of cosines. 1. the relationship between the side lengths in △abd is ( c^2 = x^2 + h^2 ) by the pythagorean theorem. 2. the relationship between the side lengths in △cbd is ( a^2 = (b - x)^2 + h^2 ) by the dropdown with options: cosine relationship, law of sines, pythagorean theorem. diagram: triangle abc with base ( b ), point d on ac, bd perpendicular to ac (height ( h )), ad = ( x ), dc = ( b - x ), ab = ( c ), bc = ( a ), bd = ( h ).

Explanation:

To determine the relationship for \(\triangle CBD\), we observe that \(\triangle CBD\) is a right triangle (since \(BD\) is perpendicular to \(AC\), forming a right angle at \(D\)). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In \(\triangle CBD\), the hypotenuse is \(a\), and the other two sides are \((b - x)\) and \(h\). So the relationship \(a^{2}=(b - x)^{2}+h^{2}\) is derived from the Pythagorean theorem. The cosine relationship and law of sines are not applicable here as they deal with non - right triangle angle - side relationships or different types of angle - side proportionalities, while this is a right - triangle side - side relationship.

Answer:

Pythagorean theorem