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In an equivalent triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.
The minimum possible area of the triangle PQR, in cm2, is
- a)144√3
- b)48√3
- c)18
- d)24
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
In an equivalent triangle PQR, side PQ is divided into four equal par...
Minimum possible area of the triangle PQR:
Given:
- Side PQ is divided into four equal parts
- Side QR is divided into six equal parts
- Side PR is divided into eight equal parts
Calculating the lengths of the sides:
Since side PQ is divided into four equal parts, each part is $\frac{1}{4}$ of PQ.
Similarly, each part of side QR is $\frac{1}{6}$ of QR, and each part of side PR is $\frac{1}{8}$ of PR.
Let the length of PQ be 4x, QR be 6y, and PR be 8z, where x, y, and z are integers.
Then, each part of PQ is x, each part of QR is y, and each part of PR is z.
Calculating the area of the triangle:
Since the triangle is equilateral, the height of the triangle can be calculated using the Pythagorean theorem.
Let the height be h.
$h^2 = x^2 - (y/2)^2 = (3y/2)^2$
$h^2 = x^2 - y^2/4 = 9y^2/4$
$4h^2 = 4x^2 - y^2 = 9y^2$
$4h^2 = 16x^2 - 4y^2 = 9y^2$
Finding the minimum possible area:
The area of the triangle PQR can be calculated using the formula:
Area = $\frac{1}{2}$ * base * height
Substitute the values of base and height in the formula and simplify to get:
Area = $\frac{1}{2}$ * 4x * h
Area = 2xh
Substitute the value of h from the calculation above:
Area = 2x * $\sqrt{\frac{16x^2 - 4y^2}{9}}$
Area = 2x * $\frac{2}{3}$y
Area = $\frac{4}{3}$xy
Since x, y, and z are integers, the minimum possible area will occur when x = 3, y = 2.
Area = $\frac{4}{3}$ * 3 * 2 = 8
Therefore, the minimum possible area of the triangle PQR is 8 cm², which is option A.
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Community Answer
In an equivalent triangle PQR, side PQ is divided into four equal par...
Let the length of side of an equilateral triangle be 24 cm. (L.C.M of 4, 6, 8 = 24)
To have minimum area of triangle PQR, (length of each sub divided part is an integer)
Area of an equilateral triangle = √3/4 a2 (where a is the length of the side)
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In an equivalent triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.The minimum possible area of the triangle PQR, in cm2, isa)144√3b)48√3c)18d)24Correct answer is option 'A'. Can you explain this answer?
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