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 dropdown with options: cosine relationship, law of sines, pythagorean theorem. (accompanying image: triangle $abc$ with $d$ on $ac$, $bd = h$ (perpendicular to $ac$), $ad = x$, $dc = b - x$, $ac = b$, $ab = c$, $bc = a$.)

Explanation:

In triangle \( \triangle ABD \), \( \angle ADB \) is a right angle (since \( BD \) is an altitude, forming a right angle with \( AC \)). 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 \( \triangle ABD \), the hypotenuse is \( c \), and the other two sides are \( x \) and \( h \). So the relationship \( c^{2}=x^{2}+h^{2} \) is derived from the Pythagorean theorem. The law of sines relates the sides and angles of a non - right triangle, and the cosine relationship is part of the law of cosines or right - triangle trigonometry but not the basis for this specific equation in a right - triangle.

Answer:

Pythagorean theorem