帮我讲解一下这道题---**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
**English Translation of 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
**Geometric Figure Description:**
* Type: Geometric figure (Triangle).
* Main Elements:
* Triangle ABC with vertices labeled A, B, C.
* Line segment DE connecting side AB and side AC. D is on AB, E is on AC. DE is described as the midsegment of △ABC, implying D is the midpoint of AB and E is the midpoint of AC, and DE is parallel to BC.
* Line segment BF originating from vertex B and extending inside the triangle, intersecting DE at F. BF is described as the angle bisector of ∠ABC.
* Point F is the intersection of BF and DE. F is located on the line segment DE.
**Given Information:**
* DE is the midsegment of △ABC.
* BF is the angle bisector of ∠ABC.
* AB = 6
* BC = 9
**Question:**
Find the length of EF.
**Options:**
A. 0.5
B. 1
C. 1.5
D. 2
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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 DE at point F. Given that AB equals 6 and BC equals 9, we need to find the length of segment EF.
First, let's establish the properties of the midsegment. Since DE is the midsegment of triangle ABC, point D is the midpoint of side AB and point E is the midpoint of side AC. This means DE is parallel to BC and has length equal to half of BC. Since BC equals 9, the length of DE is 9 divided by 2, which equals 4.5.
Now let's analyze the angle relationships. Since BF is the angle bisector of angle ABC, it divides the angle into two equal parts: angle ABF equals angle FBC. Additionally, since DE is parallel to BC, when line BF acts as a transversal, it creates alternate interior angles. This means angle DFB equals angle FBC. Therefore, angle ABF equals angle DFB.
Since we established that angle DBF equals angle DFB, triangle DBF is an isosceles triangle with two equal angles. In an isosceles triangle, the sides opposite to equal angles are also equal. Therefore, side DF equals side DB. Since AB equals 6 and D is the midpoint of AB, we have DB equals AB divided by 2, which is 3. Therefore, DF also equals 3.
Now we can find the length of EF. Since F lies on segment DE, we have DE equals DF plus EF. Substituting our known values: 4.5 equals 3 plus EF. Solving for EF: EF equals 4.5 minus 3, which equals 1.5. Therefore, the length of segment EF is 1.5 units.