find the missing side.

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle (ABC) with hypotenuse (AB = 29) and one side (AC = 23), by the Pythagorean theorem (a^{2}+b^{2}=c^{2}), where (c) is the hypotenuse and (a) and (b) are the legs of the right - triangle. Let the missing side be (a), then (a=\sqrt{AB^{2}-AC^{2}}).

Step2: Substitute values

Substitute (AB = 29) and (AC = 23) into the formula: (a=\sqrt{29^{2}-23^{2}}=\sqrt{(29 + 23)(29 - 23)}) (using the difference - of - squares formula (x^{2}-y^{2}=(x + y)(x - y))).

Step3: Calculate

First, calculate ((29 + 23)(29 - 23)=(52)\times(6)=312). Then (a=\sqrt{312}=\sqrt{4\times78}=2\sqrt{78}\approx 17.66).

Answer:

(2\sqrt{78}\approx17.66)