Triangle Folding Geometry Problem
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A piece of paper in the shape of triangle ABC is folded along segment [AD], and after folding [AC] // [DB']. $|AE| = 2 \cdot |EB'|$ and $|BD| = 3$ cm, what is $|EC|$, in cm?
This question includes visual content: The image shows two parts of a geometry problem. The top part shows a triangle ABC with a dashed line segment AD. An arrow indicates folding the triangular flap ABD along line AD. The bottom part shows the resulting shape after the fold where point B moves to B'. A new triangle AB'D is formed, and the overlap with the original base BC occurs at point E. The segment BC now contains points B, D, E, C in order. There is a blue-filled triangle A E D and a yellow-filled triangle A E C. A label '3' is placed under the segment BD. The text states |AE| = 2|EB'| and [AC] // [DB'].
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Step by Step Written Solution
Hi Bekirhan, let's solve this geometry problem together. We are looking at a folding problem involving a triangle A B C.
We are given that triangle A B C is folded along the line A D. When we fold the triangle, the point B moves to a new position, denoted as B prime.
Folding Properties
Because A D is the line of reflection, certain side lengths and angles remain equal. For instance, the length B D must be equal to the length D B prime because B prime is just the image of B.
Next, let's look at the parallel line condition. The problem states that segment A C is parallel to D B prime.
Let's identify some similar triangles created by these parallel lines.
Similarity and Parallel Lines
Notice that the line A B prime crosses the parallel lines A C and D B prime at points A, E, and B prime.
Because A C is parallel to D B prime, triangle E A C is similar to triangle E B prime D by the Angle Angle similarity theorem.
From this similarity, we can set up a ratio of corresponding sides. The ratio of E C to E D is equal to the ratio of A E to E B prime.
We are given a specific ratio in the problem: A E equals two times E B prime.
This simplifies to tell us that the length E C is twice the length E D.
Now we need to connect this to the length B D, which we know is 3 centimeters.
Using the Angle Bisector
Remember that because A D is a line of reflection for the fold, the angle B A D must equal the angle D A B prime.
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