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Let PQRSTU be a regular hexagon. The ratio of the area of the triangle PRT to that of the hexagon PQRSTU is
- a)0.3
- b)0.5
- c)1
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let PQRSTU be a regular hexagon. The ratio of the area of the triangle...
It's given that PQRSTU is a regular hexagon and O is the center of the hexagon.
If we fold TSR , PQR , TUP along lines TR, PR, PT respectively then vertices S,Q,U will overlap each other exactly at center of the hexagon.
Hence we can say that Area of hexagon PQRSTU = 2( PRT)
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Let PQRSTU be a regular hexagon. The ratio of the area of the triangle...
Solution:
Given, PQRSTU is a regular hexagon.
To find: The ratio of the area of the triangle PRT to that of the hexagon PQRSTU.
Let a be the side of the hexagon.
Area of the hexagon PQRSTU = 6 × (1/2) × a^2 × (√3/2) = 3√3a^2 sq units.
In a regular hexagon, the diagonal of one triangle is the side of the hexagon.
So, PR = a.
Let M be the midpoint of PQ, then PM = MQ = (a/2).
Also, angle TPM = angle TQM = 60°.
Therefore, triangle TPM and TQM are congruent.
So, TP = TQ = a/2.
Then, the area of triangle PRT = (1/2) × PR × TP = (1/2) × a × a/2 = a^2/4 sq units.
So, the required ratio = area of triangle PRT/ area of hexagon PQRSTU
= (a^2/4)/(3√3a^2)
= 1/(4√3)
= (√3/12)
= 0.5 (approx)
Hence, the correct option is (B) 0.5.
Given, PQRSTU is a regular hexagon.
To find: The ratio of the area of the triangle PRT to that of the hexagon PQRSTU.
Let a be the side of the hexagon.
Area of the hexagon PQRSTU = 6 × (1/2) × a^2 × (√3/2) = 3√3a^2 sq units.
In a regular hexagon, the diagonal of one triangle is the side of the hexagon.
So, PR = a.
Let M be the midpoint of PQ, then PM = MQ = (a/2).
Also, angle TPM = angle TQM = 60°.
Therefore, triangle TPM and TQM are congruent.
So, TP = TQ = a/2.
Then, the area of triangle PRT = (1/2) × PR × TP = (1/2) × a × a/2 = a^2/4 sq units.
So, the required ratio = area of triangle PRT/ area of hexagon PQRSTU
= (a^2/4)/(3√3a^2)
= 1/(4√3)
= (√3/12)
= 0.5 (approx)
Hence, the correct option is (B) 0.5.
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Let PQRSTU be a regular hexagon. The ratio of the area of the triangle PRT to that of the hexagon PQRSTU isa)0.3b)0.5c)1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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