## OG 2020: Question No. 325

In the figure above, the vertices of ΔOPQ and ΔQRS have coordinates as indicated. Do ΔOPQ and ΔQRS have equal areas?

- b = 2a
- d = 2c

Source | OG 2020 |

Type | Data Sufficiency |

Topic | Geometry |

Sub-Topic | Triangles / Coordinate Geometry |

Difficulty | Medium |

### Solution

__Steps 1 & 2: Understand Question and Draw Inferences__

__Steps 1 & 2: Understand Question and Draw Inferences__

In this question, we are given

- A diagram, comprising of ΔOPQ and ΔQRS, along with the coordinates of their vertices, as shown.

We need to determine

- Whether ΔOPQ and ΔQRS have equal areas or not.

We know the area of a triangle is given by ½ * base * height.

- Area of ΔOPQ = ½ * OQ * height = ½ * c * 3 = 3c/2
- Area of ΔQRS = ½ * QS * height = ½ * (d – c) * 3 = 3(d-c)/2

Now, if the triangles have equal area, then

- 3c/2 = 3(d-c)/2

Or, c = d – c

Or, d = 2c

Hence, we can say that if we know whether d = 2c, we can term the statement sufficient.

With this understanding, let us now analyse the individual statements.

__Step 3: Analyse Statement 1__

__Step 3: Analyse Statement 1__

As per the information given in statement 1, b = 2a.

- However, from this statement, we cannot say whether d = 2c or not.

Hence, statement 1 is not sufficient to answer the question.

__Step 4: Analyse Statement 2__

__Step 4: Analyse Statement 2__

As per the information given in statement 2, d = 2c.

- Since d = 2c, we can conclude that the areas of both triangles are equal.

Hence, statement 2 is sufficient to answer the question.

__Step 5: Combine Both Statements Together (If Needed)__

__Step 5: Combine Both Statements Together (If Needed)__

Since we can determine the answer from statement 2 individually, this step is not required.

Hence, the correct answer choice is option B.

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