Bunuel wrote:
In the figure above, if AB is parallel to DE, DC = EC, what is the perimeter of triangle DEC?
(1) BE is 5/7 of BC
(2) AC = 13 and AB = 7
Solution
Step 1: Analyse Question Stem
Before moving on to the solution of this problem, we need to know a few properties of triangles.
• Look at the diagram given,
o If \(DE\) is parallel to \(AB\), then triangle \(ABC\) is similar to triangle \(DEC\).
Thus, \(\frac{EC}{BC} = \frac{DC}{AC}=\frac{DE}{AB}\)
• Now, by using this property let’s solve the given problem.
o \(AB\) is parallel to \(DE\)
Triangle \(ABC\) is similar to triangle \(DEC\).
• Thus, \(\frac{EC}{BC}=\frac{DC}{AC}=\frac{DE}{AB}\)
It is also given that \(DC = EC.\)
\(DE = 2\) units
We need to find the perimeter of triangle \(DEC\).
• \(DE+EC+DC = 2 + EC + DC= 2+2DC\) [it is given that \(DC = EC\)]
.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: \(BE = \frac{5}{7}*BC\)
• \(\frac{BE}{BC} = \frac{5}{7}\)
o \(1-\frac{BE}{BC} = 1 – \frac{5}{7}\) [Subtracting both the sides from \(1\)]
o \(\frac{(BC-BE)}{BC} = \frac{2}{7} \)
o EC/BC = 2/7, [BE+EC = BC]
• It means \(\frac{EC}{BC} = \frac{DC}{AC}=\frac{DE}{AB} = \frac{2}{7}\).
• With this information we can find the ratio of sides, but we cannot find the value of \(DC\) or \(EC\).
Hence, statement 1 is not sufficient, we can eliminate answer options A and D.
Statement 2: \(AC = 13\) and \(AB = 7\).
• \(\frac{DC}{AC} = \frac{DE}{AB}\)
o \(\frac{DC}{13} = \frac{2}{7}\)
o \(DC = \frac{26}{7}\)
• Perimeter of triangle DEC = \(2 + 2*\frac{26}{7}\)
Hence, statement 2 is sufficient, the correct answer is
Option B.
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