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In an equilateral triangle PQR, side PQ is divided in 4 equal parts, side QR is divided into 6 equal parts and PR is divided into 8 equal parts. The length of each sub-divided part in cm is an integer. The minimum area of triangle PQR possible, in cm2, is
- a)18
- b)144√3
- c)48√3
- d)24
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
In an equilateral triangle PQR, side PQ is divided in 4 equal parts, s...
Let's assume the length of each sub-divided part on side PQ is x cm.
Therefore, the total length of side PQ = 4x cm.
Similarly, let's assume the length of each sub-divided part on side QR is y cm.
Therefore, the total length of side QR = 6y cm.
And finally, let's assume the length of each sub-divided part on side PR is z cm.
Therefore, the total length of side PR = 8z cm.
Since triangle PQR is equilateral, all sides are equal in length.
So, 4x = 6y = 8z.
To find the minimum area of the triangle, we need to find the minimum values of x, y, and z that satisfy the equation 4x = 6y = 8z.
The smallest common multiple of 4, 6, and 8 is 24. So, let's assume x = 24, y = 24/4 = 6, and z = 24/3 = 8.
Now, we can calculate the length of each side of the triangle:
Side PQ = 4x = 4 * 24 = 96 cm
Side QR = 6y = 6 * 6 = 36 cm
Side PR = 8z = 8 * 8 = 64 cm
Using Heron's formula, we can calculate the area of the triangle:
s = (96 + 36 + 64) / 2 = 98 cm (where s is the semi-perimeter of the triangle)
Area = √(s(s - Side PQ)(s - Side QR)(s - Side PR))
= √(98(98 - 96)(98 - 36)(98 - 64))
= √(98(2)(62)(34))
= √(98 * 2 * 62 * 34)
= √(4050144)
= 2016 cm²
Therefore, the minimum area of triangle PQR possible is 2016 cm², which is not one of the given options.
Therefore, the total length of side PQ = 4x cm.
Similarly, let's assume the length of each sub-divided part on side QR is y cm.
Therefore, the total length of side QR = 6y cm.
And finally, let's assume the length of each sub-divided part on side PR is z cm.
Therefore, the total length of side PR = 8z cm.
Since triangle PQR is equilateral, all sides are equal in length.
So, 4x = 6y = 8z.
To find the minimum area of the triangle, we need to find the minimum values of x, y, and z that satisfy the equation 4x = 6y = 8z.
The smallest common multiple of 4, 6, and 8 is 24. So, let's assume x = 24, y = 24/4 = 6, and z = 24/3 = 8.
Now, we can calculate the length of each side of the triangle:
Side PQ = 4x = 4 * 24 = 96 cm
Side QR = 6y = 6 * 6 = 36 cm
Side PR = 8z = 8 * 8 = 64 cm
Using Heron's formula, we can calculate the area of the triangle:
s = (96 + 36 + 64) / 2 = 98 cm (where s is the semi-perimeter of the triangle)
Area = √(s(s - Side PQ)(s - Side QR)(s - Side PR))
= √(98(98 - 96)(98 - 36)(98 - 64))
= √(98(2)(62)(34))
= √(98 * 2 * 62 * 34)
= √(4050144)
= 2016 cm²
Therefore, the minimum area of triangle PQR possible is 2016 cm², which is not one of the given options.
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Community Answer
In an equilateral triangle PQR, side PQ is divided in 4 equal parts, s...
For
to be integer, a must be LCM of 4, 6 and 8. So a = 24 
Hence, the correct option is (B).
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In an equilateral triangle PQR, side PQ is divided in 4 equal parts, side QR is divided into 6 equal parts and PR is divided into 8 equal parts. The length of each sub-divided part in cm is an integer. The minimum area of triangle PQR possible, in cm2, is a)18b)144√3c)48√3d)24Correct answer is option 'B'. Can you explain this answer?
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