Geometry Paper Folding Problem
Published:
A piece of paper in the shape of triangle ABC is folded along segment $[AD]$, and after folding $[AC] // [DB']$.
$|AE| = 2 · |EB'|$ and $|BD| = 3 \text{ cm}$,
what is $|EC|$, in cm?
A) 5
B) 4
C) 6
D) $\frac{9}{2}$
E) $\frac{11}{2}$
This question includes visual content: A triangle ABC is illustrated. Part of it is folded along segment AD, resulting in a new vertex B'. The triangle is divided into a teal triangle (ADE) and a tan triangle (ADC). Segment BD is horizontal, and segment DB' is at an angle. The text specifies |AE| = 2|EB'| and |BD| = 3 cm, and mentions that after folding, [AC] is parallel to [DB'].
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Step by Step Written Solution
Hi Bengisu, let's solve this geometry folding problem together.
Geometry: Folding Triangle ABC
When we fold triangle A B D along the segment A D to get triangle A B prime D, several properties stay the same.
Key Property of Folding:
Folding creates congruent triangles and angle bisectors.
Because the folding line is A D, the angle B A D is equal to the angle B prime A D. This means A D is the angle bisector of angle B A B prime.
The problem states that after folding, the segment A C is parallel to D B prime. Let's mark this key information.
Since A C and D B prime are parallel, we can identify alternate interior angles. Specifically, angle C A D is equal to angle A D B prime. Let's call this measure beta.
Let's look at the triangle properties we have gathered so far.
Analyzing the Triangle A E C
Since B D is 3 and it was folded onto D B prime, the length of D B prime is also 3 centimeters.
Now consider the similarity between triangle D B prime E and triangle C A E because A C is parallel to D B prime.
By similarity (AA Criterion):
Because these triangles are similar, the ratio of their corresponding sides must be equal. This includes the ratio of D B prime to A C as well as the segments on the transversal lines.
The problem gives us the ratio for the segments on the transversal: A E equals two times E B prime.
Substituting this back into our similarity ratio, we find that D B prime over A C must also be one over two.
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