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A square PQRS has an equilateral triangle PTO inscribed as shown:
What is the ratio of AΔPQT to AΔTRU?
- a)1 : 3
- b)1 : √3
- c)1 : √2
- d)1 : 2
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
A square PQRS has an equilateral triangle PTO inscribed as shown:What ...
Let PQ, a side of equilateral triangle be b
By symmetry QT=ST=z (say)
=) a^2 + z^2 – 2az = 2az (Please note how the solution is being managed here. You must always be aware of what you are looking for. Here, as equation -℗ we are looking for (a-z)2 in terms of az)
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A square PQRS has an equilateral triangle PTO inscribed as shown:What ...
Let the side length of the square be x. Then the side length of the equilateral triangle is also x.
We can use the Pythagorean theorem to find the length of PO:
PO² = PT² + OT²
PO² = x² + (x/2)²
PO² = 5x²/4
PO = x√5/2
Similarly, we can find the length of PQ:
PQ² = PO² + OQ²
PQ² = (x√5/2)² + (x/2)²
PQ² = 9x²/4
PQ = 3x/2
Now we can find the area of the triangle PTO:
Area(PTO) = (1/2) * PT * OT
Area(PTO) = (1/2) * x * (x/2)
Area(PTO) = x²/4
And we can find the area of the square PQRS:
Area(PQRS) = x²
The ratio of the area of the triangle to the area of the square is:
Area(PTO) / Area(PQRS) = (x²/4) / x²
Area(PTO) / Area(PQRS) = 1/4
Therefore, the ratio of the area of the triangle to the area of the square is 1:4.
We can use the Pythagorean theorem to find the length of PO:
PO² = PT² + OT²
PO² = x² + (x/2)²
PO² = 5x²/4
PO = x√5/2
Similarly, we can find the length of PQ:
PQ² = PO² + OQ²
PQ² = (x√5/2)² + (x/2)²
PQ² = 9x²/4
PQ = 3x/2
Now we can find the area of the triangle PTO:
Area(PTO) = (1/2) * PT * OT
Area(PTO) = (1/2) * x * (x/2)
Area(PTO) = x²/4
And we can find the area of the square PQRS:
Area(PQRS) = x²
The ratio of the area of the triangle to the area of the square is:
Area(PTO) / Area(PQRS) = (x²/4) / x²
Area(PTO) / Area(PQRS) = 1/4
Therefore, the ratio of the area of the triangle to the area of the square is 1:4.
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