Question: In isosceles $\triangle ABC$ with $BM$ as median, and $BM : AC = 1 : 2$. Which of the following is the correct statement and reason? (A) $\triangle ABC$ is isosceles by Median to hypotenuse Theorem. (B) $\triangle ABC$ is isosceles by Median to hypotenuse Theorem Converse. (C) $\triangle ABC$ is right by Median to hypotenuse Theorem (D) $\triangle ABC$ is right by Median to hypotenuse Theorem Converse. Graph: top-right, an isosceles-looking triangle $\triangle ABC$ with $B$ at the top, $A$ and $C$ at the bottom endpoints, $M$ on $AC$ as the midpoint, and segment $BM$ drawn vertically from $B$ to $M$; the text states $BM : AC = 1 : 2$. find $BM$

1. **Problem Statement:** We have an isosceles triangle $\triangle ABC$ with $BM$ as the median to side $AC$. The ratio $BM : AC = 1 : 2$. We need to determine which statement about the triangle is correct based on the Median to Hypotenuse Theorem. 2. **Recall the Median to Hypotenuse Theorem:** In a right triangle, the median drawn to the hypotenuse is half the length of the hypotenuse. That is, if $\triangle ABC$ is right angled at $B$, and $M$ is the midpoint of hypotenuse $AC$, then: $$BM = \frac{1}{2} AC$$ 3. **Given Ratio:** $BM : AC = 1 : 2$ means: $$\frac{BM}{AC} = \frac{1}{2}$$ This matches exactly the condition of the Median to Hypotenuse Theorem. 4. **Interpretation:** Since $BM$ is half of $AC$, and $M$ is midpoint of $AC$, by the Median to Hypotenuse Theorem, $\triangle ABC$ must be a right triangle with the right angle at $B$. 5. **Conclusion:** The correct statement is: (C) $\triangle ABC$ is right by Median to hypotenuse Theorem. --- **Final answer:** (C) $\triangle ABC$ is right by Median to hypotenuse Theorem.