Final answer:
The statement 'BC = CD' is true when AC is a median in a triangle; it bisects the opposite side into two equal parts. However, angle equality, such as in statements 'BAC = DAC' or '∠B = ∠D', is not guaranteed by the existence of a median alone. The correct answer is option B .
Step-by-step explanation:
When we talk about a median in geometry, specifically regarding a triangle, we refer to a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Given this definition, there are a few properties that a median has which are useful to consider when answering the question "The segment AC is a median. Which is a true statement?".
Consider a triangle ABC where AC is a median, which means it connects vertex A to midpoint C on side AB. Here are some properties with respect to the median:
- B) BC = CD: This is true. Since AC is a median, it divides side AB into two equal segments, BC and CD.
- C) BAC = DAC: This is false. Even though AC is a median, it does not always guarantee that angles BAC and DAC are equal.
- D) ∠B = ∠D: This is true if the triangle is isosceles and AC is the median to the base.
In conclusion, the property associated with the median in a triangle is equal subdivision of the side it bisects, which makes option B the only universally true statement in a triangle with a median. In a triangle, a median is a line segment drawn from a vertex to the midpoint of the opposite side. Let's analyze each statement:
A) MACD = 90: This statement is false. In a triangle, the median does not necessarily intersect the triangle at a right angle.
B) BC = CD: This statement is true. The median divides the opposite side into two equal segments, so BC is equal to CD.
C) BAC = DAC: This statement is true. The angles opposite to equal sides in a triangle are equal, so BAC is equal to DAC.
D) ∠B = ∠D: This statement is false. The angles opposite to equal sides in a triangle are equal, but the median does not guarantee that ∠B is equal to ∠D.
Therefore, the correct answer is B) BC = CD.