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ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5?
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ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,B...
Solution:
Step 1: Finding the Area of the Hexagon
- We know that the area of a regular hexagon with side 'a' is 3√3/2 * a^2.
- In this case, the side of the hexagon is given as AB. Let the side of the hexagon be 'x'.
- Since ABCDEF is a regular hexagon, AB = BC = CD = DE = EF = FA = x.
- Therefore, the area of the hexagon = 3√3/2 * x^2.
Step 2: Finding the Area of Triangle PQR
- Let's draw a diagram to understand the position of P, Q, and R.
- Since P is the mid-point of DE, we know that DP = PE = x/2.
- Similarly, we can find that AQ = QF = x/2 and BR = RC = x/2.
- Now, we can draw lines PQ and QR to form triangle PQR.
- Let's extend DP to meet AB at S. Since ABCDEF is a regular hexagon, AS = SB = x/2 * √3.
- Similarly, we can find that CR = RD = x/2 * √3.
- Now, we know that triangle ADS and triangle CRS are congruent (by SAS congruence).
- Therefore, area of triangle PQR = 2 * area of triangle ADS.
- Area of triangle ADS = 1/2 * AD * DS = 1/2 * (x/2) * (x/2 * √3) = x^2/8 * √3.
- Therefore, area of triangle PQR = 2 * (x^2/8 * √3) = x^2/4 * √3.
Step 3: Finding the Ratio of Areas
- Ratio of areas of triangle PQR to hexagon = (x^2/4 * √3) / (3√3/2 * x^2) = 1/6.
- Therefore, the ratio of areas of triangle PQR to hexagon is 1 : 6.
Step 4: Simplifying the Ratio
- We can simplify the ratio by dividing both sides by 2.
- Therefore, the ratio of areas of triangle PQR to hexagon is 1/2 : 3.
- Therefore, the correct answer is option (c) 3:8.
Community Answer
ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,B...
1:3
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ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5?
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ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5?.
ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCDEF is a regular hexagon. P, Q and R are mid-points of DE,AF and ,BC respectively. Ratio of areas of triangle PQR to hexagon is a) 1 : 2 b) 1 : 3 c) 3:8 d) 3:5?.
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