1. **Problem Statement:** We are given a triangle ABC with a right angle at C and height AC. We need to analyze the given statements and explain why option C is false. 2. **Understanding the triangle and notation:** - $m AC$ denotes the length of segment AC (height). - $m AD$, $m BC$, etc., denote lengths of respective segments. - Angles are denoted by $\angle$ followed by points, e.g., $\angle ADC$. 3. **Reviewing each option:** **A) $\sin \angle ADC = \frac{m AC}{m AD}$** - In right triangle ADC, $\sin \angle ADC = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{m AC}{m AD}$. - This is true by definition of sine. **B) $m AC = \frac{m AD}{2}$** - This is a specific length relation that depends on the triangle's dimensions. - Given the height and base, this can be true if AC is half of AD. **C) $\tan \angle ABC = \frac{m BC}{m AC}$** - $\tan \angle ABC = \frac{\text{opposite side to } \angle ABC}{\text{adjacent side to } \angle ABC}$. - At vertex B, opposite side is AC, adjacent side is BC. - So $\tan \angle ABC = \frac{m AC}{m BC}$, not $\frac{m BC}{m AC}$. - Therefore, option C is false. **D) $\frac{m BC}{\sin \angle BAC} = \frac{m AC}{\sin \angle ABC}$** - This is a form of the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$. - This relation holds true in any triangle. 4. **Summary:** - Option C incorrectly swaps the opposite and adjacent sides in the tangent ratio. - The correct formula is $\tan \angle ABC = \frac{m AC}{m BC}$. **Final answer:** Option C is false because it incorrectly states the tangent ratio for $\angle ABC$.