Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe

Kudos for a diagram of this problem

Solution:

Steps 1 & 2: Understand Question and Draw Inferences

Given:

• Isosceles ΔABC

o AB = AC
o So, perpendicular AD will bisect side BC.
o If we assume ∠ ABC to be \(x^o\), the different ∠s in the figure can be represented as below:

To find: Perimeter of ΔABC

• =AB + BC + CA
• = 2AB + BC

So, to find the perimeter, we need to know AC and BC
Knowing the ∠s may help us (because we may then employ trigonometric ratios)

Step 3: Analyze Statement 1 independently

(1) ∠BAD = 2∠ACD

90°- x° = 2x°
⇒ 3x° = 90°
⇒ x° = 30°

But we don’t know the magnitude of any side of the triangle. So, we cannot yet use trigonometric ratios to find the sides of the triangle.

Not sufficient.

Step 4: Analyze Statement 2 independently

(2) The perimeter of triangle ADB is 15 + 5√3

\(\begin{array}{l}\mathrm{AB}\;+\;\mathrm{BD}\;+\;\mathrm{AD}\;=\;15\;+\;5\surd3\\\Rightarrow\mathrm{AB}\;+\frac{\mathrm{BC}}2\;+\;\mathrm{AD}\;=\;15\;+\;5\sqrt3\\\Rightarrow2\mathrm{AB}+\mathrm{BC}=30+10\sqrt3-2\mathrm{AD}\end{array}\)

The value of 2AB + BC depends on the value of AD.
Since we do not know the value of AD, we cannot find the required value

Not sufficient.

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1:



• From Statement 2: \(2\mathrm{AB}+\mathrm{BC}=30+10\sqrt3-2\mathrm{AD}\) . . . (I)
• In right ΔADB
\(\frac{\mathrm{AD}}{\mathrm{AB}}=\sin⁡30^\circ=\frac12\)
⇒ \(\mathrm{AD}=\frac{\mathrm{AB}}2\) . . . (II)

\(\begin{array}{l}\frac{\mathrm{BD}}{\mathrm{AB}}=\cos⁡30^\circ=\frac{\sqrt3}2\\\Rightarrow\mathrm{BD}=\frac{\mathrm{BC}}2=\frac{(\sqrt3)\mathrm{AB}}2\\\Rightarrow\mathrm{BC}=(\sqrt3)\mathrm{AB}\end{array}\)
• So, \(2\mathrm{AB}+(\sqrt3)\mathrm{AB}=30+10\sqrt3-2\ast\frac{\mathrm{AB}}2\)

o A linear equation in AB. So, we’ll get a unique value of AB

• Using Equation II, a unique value of AD will be obtained
• Using Equation I, the value of 2AB + BC will be obtained

Sufficient.

Answer: Option C

Thanks,
Saquib
Quant Expert
e-GMAT

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