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# In the figure above, the vertices of ΔOPQ and ΔQRS have coordinates as indicated. Do ΔOPQ and ΔQRS have equal areas? – OG 2020 Question #325 with Solution

## 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?

1. b = 2a
2. 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

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

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

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)

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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