In the figure above, B is on AC, D is on AE, AB has the same length as

in the triangles ABD and ACE
\(\angle ABD\) = \(\angle ACE\)
\(\angle A\) is common
\(\angle ADB\) = \(\angle AEC\)

Hence the two triangles are similar.
=> ratio of AB and AC = Ratio of DB and EC
(1) When EC = 6 we can calculate DB by the above
SUFFICIENT

(2)DE = 5
There is no way to know DB from this.
INSUFFICIENT

Hence A IMO

The two triangles are similar triangle. From the base the ratio of their sides is 2 :1.

Statement 1 :knowing that EC is 6, we can tell that DB is 3.
Statement 2 : we can tell that DE is 5 and hence AD is 5 but that does not give us the value of DB.

answer is A

AB=BC, so B is the mid-point of AC.

Angle ABD = Angle ACE, so we can conclude that line DB is parallel to line CE as stated angles are corresponding angles to these parallel lines.

We can simple apply mid-point theorem and conclude that DB is half of CE.

St1 provides the value of CE whereas St2 does not.

Hence, Ans A

Answer: Option A

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No calculations required.

In the figure above, B is on AC, D is on AE, AB has the same length as BC, and ∠ABD∠ABD has the same measure as ∠ACE∠ACE. What is the length of DB ?

(1) The length of EC is 6.
The stem tells us that EC and DB are parallel to one another and that DB is the mid-segment of EC. Hence, DB = 3.
Sufficient.

(2) The length of DE is 5.
There's nothing else we can determine from this.
Insufficient.

A
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Step 1: Analyse Question Stem

Invaluable question data given in this question, including the figure.

Triangles ABD and ACE share a common angle A.
Angle ABD = Angle ACE.
Two angles of triangle ABD are equal to two angles of triangle ACE. Therefore, angle ADB MUST be equal to angle AEC and so, the two triangles must be similar.

Since the two triangles are similar, the corresponding sides are in proportion.

\(\frac{AB }{ AC}\) = \(\frac{AD }{ AE}\) = \(\frac{BD }{ CE}\)

Let us note that AC = AB + BC; also AB = BC. Therefore, AC = 2 AB or \(\frac{AB }{ AC }\)= \(\frac{1 }{ 2}\).

Therefore, \(\frac{AD }{ AE}\) = \(\frac{BD }{ CE}\) = \(\frac{1 }{ 2}\)

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: The length of EC is 6.

Using the ratio from the analysis of the question, \(\frac{BD }{ CE}\) = \(\frac{1 }{ 2}\)
Therefore, BD = 3

The data in statement 1 is sufficient to find a unique value for the length of DB
Statement 1 alone is sufficient. Answer options B, C and E can be eliminated.

Statement 2: The length of DE is 5

From the ratio in the analysis of the question, we know that \(\frac{AD }{ AE}\) = \(\frac{1 }{ 2}\)
Also, AE = AD + DE and DE = 5.

\(\frac{AD }{ AD + 5}\) = \(\frac{1 }{ 2}\)

Simplifying and solving, we have AD = 5.
Knowing AD is not sufficient to find the value of EC and hence DB.

The data in statement 2 is insufficient to find a unique value for the length of DB
Statement 2 alone is insufficient. Answer option D can be eliminated.

The correct answer option is A.
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