Solve this program---**Question Number:** 5
**Question Stem:**
如图, DE 是△ABC 的中位线, ∠ABC 的角平分线交 DE 于点 F, AB = 6, BC = 9, 则 EF 的长为
**Translated Question Stem:**
As shown in the figure, DE is the midsegment of △ABC, the angle bisector of ∠ABC intersects DE at point F, AB = 6, BC = 9, then the length of EF is
**Options:**
A. 0.5
B. 1
C. 1.5
D. 2
**Chart/Diagram Description:**
* **Type:** Geometric figure (Triangle)
* **Main Elements:**
* A triangle labeled ABC with vertices A, B, and C.
* A line segment DE connecting side AB and side AC. Point D is on AB, and point E is on AC.
* A line segment BF starting from vertex B and intersecting DE at point F.
* **Labels:** Vertices A, B, C are labeled. Points D, E, F are labeled.
* **Relative Position and Direction:**
* Points D and E are on sides AB and AC respectively.
* Line segment DE is inside the triangle.
* Point F is on the line segment DE.
* Line segment BF passes through the interior of the triangle and intersects DE.
* The line segment BC forms the base of the triangle in the diagram's orientation.
* **Implied Information from Question Stem:**
* DE is the midsegment of △ABC, meaning D is the midpoint of AB and E is the midpoint of AC. This implies DE || BC and DE = 1/2 * BC.
* BF is the angle bisector of ∠ABC.
* AB = 6, BC = 9.
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We have triangle ABC where DE is the midsegment connecting the midpoints of sides AB and AC. The angle bisector of angle ABC intersects the midsegment DE at point F. We're given that AB equals 6 and BC equals 9, and we need to find the length of segment EF.
Since DE is the midsegment of triangle ABC, we know that D is the midpoint of AB and E is the midpoint of AC. This means BD equals half of AB, which is 3. The midsegment DE is parallel to the base BC and has length equal to half of BC, so DE equals 4.5.
Now let's analyze the angle bisector. Since BF bisects angle ABC, we have angle ABF equals angle FBC. Because DE is parallel to BC, and BF is a transversal, the alternate interior angles are equal, so angle DFB equals angle FBC. This means angle ABF equals angle DFB.
Since angle ABF equals angle DFB, triangle BDF is isosceles with BD equal to DF. We know BD equals 3, so DF also equals 3. Since the total length DE is 4.5 and DF is 3, we can find EF by subtracting: EF equals DE minus DF, which is 4.5 minus 3, equals 1.5.
Let's summarize our solution. We found that DE is the midsegment with length 4.5, D is the midpoint so BD equals 3, the angle bisector creates an isosceles triangle BDF making DF equal to 3, and therefore EF equals DE minus DF, which is 1.5. The answer is C.