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655-705 (Hard)|   Geometry|      
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Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe Updated on: 07 Aug 2018, 04:36
Originally posted by EgmatQuantExpert on 28 Feb 2017, 04:03.
Last edited by EgmatQuantExpert on 07 Aug 2018, 04:36, edited 1 time in total.
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C
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55% (hard)

Question Stats:

65% (02:33) correct 35% (02:02) wrong based on 131 sessions

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Q. Triangle ABC is an isosceles triangle with AB = AC and AD is the perpendicular dropped from vertex A on the side BC. What is the perimeter of triangle ABC?

    (1) \(∠BAD = 2∠ACD\)
    (2) The perimeter of triangle ADB is \(15 + 5√3\)


Answer Choices :

    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
    C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
    D. EACH statement ALONE is sufficient to answer the question asked.
    E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe Updated on: 27 Mar 2017, 05:20
Originally posted by EgmatQuantExpert on 28 Feb 2017, 04:04.
Last edited by EgmatQuantExpert on 27 Mar 2017, 05:20, edited 1 time in total.
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The official solution has been posted. Looking forward to a healthy discussion..:)
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Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe 08 Mar 2017, 19:47
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EgmatQuantExpert
Q. Triangle ABC is an isosceles triangle with AB = AC and AD is the perpendicular dropped from vertex A on the side BC. What is the perimeter of triangle ABC?

    (1) \(∠BAD = 2∠ACD\)
    (2) The perimeter of triangle ADB is \(15 + 5√3\)


Answer Choices :

    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
    C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
    D. EACH statement ALONE is sufficient to answer the question asked.
    E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Thanks,
Saquib
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(1) \(∠BAD = 2∠ACD\)

From the above statement, we conclude that ∠BAD = 60 and ∠ABD = 30. Since the triangle ADB is a right triangle, the ratio should be 1: √3 : 2. However, no lengths are defined in this statement. Insufficient.

2) The perimeter of triangle ADB is \(15 + 5√3\)[/list]
No angles are defined. Only the sum of sides, AD + DB + AB = 15 + 5√3. Clearly insufficient.

Combining the two, we get the ratio of lengths and equate it total lengths.

x + √3x + 2x = 15 + 5√3 => x = 5. This should be sufficient to figure out the perimeter.
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Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe 13 Mar 2017, 15:08
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Triangle ABC is an isosceles triangle with AB = AC and AD is the perpe 27 Mar 2017, 05:20
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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

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