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655-705 (Hard)|   Geometry|      
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If in triangle ABC angle ABC is the largest and point D lies 01 Nov 2010, 11:25
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57% (01:40) correct 43% (02:07) wrong based on 178 sessions

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If, in triangle \(ABC\), angle \(ABC\) is the largest and point \(D\) lies on segment \(AC\), is the area of triangle \(ABD\) larger than that of triangle \(DBC\)?

(1) \(AD < DC\)

(2) \(AB < BC\)


M18-04

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If in triangle ABC angle ABC is the largest and point D lies 03 Mar 2020, 04:04
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If, in triangle \(ABC\), angle \(ABC\) is the largest and point \(D\) lies on segment \(AC\), is the area of triangle \(ABD\) larger than that of triangle \(DBC\)?

Consider the diagram below:



Notice that \(BE\) is the height of triangle \(ABC\). Now, the area of triangle \(ABD\) is \(\frac{1}{2}*height*base = \frac{1}{2}*BE*AD\), and the area of triangle \(DBC\) is \(\frac{1}{2}*height*base=\frac{1}{2}*BE*DC\). So, we can see that the area of triangle \(ABD\) will be greater than the area of triangle \(DBC\) if \(AD\) is greater than \(DC\).

(1) \(AD \lt DC\). Sufficient.

(2) \(AB \lt BC\). Not sufficient.


Answer: A
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If in triangle ABC angle ABC is the largest and point D lies 01 Nov 2010, 16:25
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If in triangle \(ABC\) angle \(ABC\) is the largest and point \(D\) lies on segment \(AC\) , is the area of triangle \(ABD\) larger than that of triangle \(DBC\) ?

1. \(AD < DC\)
2. \(AB < BC\)
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The area of a triangle is 0.5 x base x height

Consider the triangles ABD and DBC. The perpendicular height of vertex B from the side AD is the same as that from side DC (the same as the height from B to AC in the bigger triangle).
Hence the areas are 0.5 x (Height from B to AC) x AD & 0.5 * (Height from B to AC) * DC

(1) Knowing AD < DC is sufficient to compare the areas of the triangles using the above formula

(2) Knowing AB<BC does not matter much in comparing areas. (Imagine that AB<BC, now in the two cases that AD<DC & AD>DC ... the area of one triangle can be bigger or smaller than the other using the above formula). Insufficient

Answer : A
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If in triangle ABC angle ABC is the largest and point D lies 03 Mar 2020, 04:53
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to find the area we need to know about base and height.it is not changed in this question so it will be same.
therefore we need to find whose base is greater.
option a gives me that.
so A
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If in triangle ABC angle ABC is the largest and point D lies 04 Mar 2020, 00:02
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Bunuel
In a triangle ABC point D lies on segment AC. Is the area of triangle ABD greater than the area of triangle BDC?

(1) DC > AD
(2) BC > AB

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Solution


Step 1: Analyse Question Stem


    • ABC is a triangle and D lies on side AC.
    • We need to find if area of triangle ABD is greater than the area of triangle BDC.
    • Based on the above information let’s draw the below diagram:
      o In this diagram, BE is the perpendicular drawn from B to AC

    • Now, area of triangle \(ABD = \frac{1}{2}* base* height = \frac{1}{2}*AD*BE \)
    • Also, area of triangle \(BDC = \frac{1}{2}*base*height = \frac{1}{2}*CD*BE\)
    • we need to find if area of triangle ABD \(>\) area of triangle BDC
      o Or, \(\frac{1}{2}*AD*BE > \frac{1}{2}*CD*BE ⟹ AD > CD\)
Therefore, we need to find out if \(AD > CD.\)

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


Statement 1: DC > AD
    • According to this statement: \(AD < CD\)
    •Thus, we can infer that area of triangle ABD \(<\) area of triangle BDC
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.
Statement 2: BC > AB
    • From this statement, we can’t find anything about AD and CD.
    • So, let’s consider the
      o area of triangle \(ABD = \frac{1}{2}* base* height = \frac{1}{2}*AB*GD\)
      o and, area of triangle \(BDC = \frac{1}{2}*base*height = \frac{1}{2}*BC*DH\)
    • We know that \(BC > AB\), however we don’t which one is greater between DG and DH.
    • Thus, even after considering this case too, we can’t decidedly say whether area of triangle ABD is greater than area of triangle BDC or not.

Hence, statement 2 is not sufficient.
Thus, the correct answer is Option A.
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If in triangle ABC angle ABC is the largest and point D lies 01 Jun 2023, 23:07
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