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In a regular pentagon PQRST, what is the ratio of the area of triangle PRS to the area of the pentagon PQRST?
- a)1 / (sin 72 + 4)
- b)1 / (cos 72 + 4)
- c)1 / (4 cos 72 + 1)
- d)1 / (4 sin 72 + 1)
- e)none of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a regular pentagon PQRST, what is the ratio of the area of triangle...
Each angle of a regular pentagon is 108 degrees. In triangle PTS, angle T = 108 degrees, and PT = TS. So, angle TPS = angle TSP = 36 degrees.
Similarly, angle QPR = 36 degrees. So, angle SPR = 108 – (36 + 36) = 36 degrees
Also, PS = PR
Area of triangle PSR = ½ * PS * PR * sin 36
Area of the pentagon = Sum of areas of the three triangles = Area of triangle PTS + Area of triangle PSR + Area of triangle PQR
= ½ * PT * PS * sin (angle TPS) + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin (angle QPR)
= ½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36
Required ratio = ½ * PS * PR * sin 36 / (½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36)
= PR / (PT + PR + PQ)
PT = PQ = SR
SR = 2 * PR cos (angle PSR) = 2 * PR * cos 72
Ratio = PR / (2 * PR cos 72 + PR + 2 * PR cos 72) = 1 / (4 cos 72 + 1)
Similarly, angle QPR = 36 degrees. So, angle SPR = 108 – (36 + 36) = 36 degrees
Also, PS = PR
Area of triangle PSR = ½ * PS * PR * sin 36
Area of the pentagon = Sum of areas of the three triangles = Area of triangle PTS + Area of triangle PSR + Area of triangle PQR
= ½ * PT * PS * sin (angle TPS) + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin (angle QPR)
= ½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36
Required ratio = ½ * PS * PR * sin 36 / (½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36)
= PR / (PT + PR + PQ)
PT = PQ = SR
SR = 2 * PR cos (angle PSR) = 2 * PR * cos 72
Ratio = PR / (2 * PR cos 72 + PR + 2 * PR cos 72) = 1 / (4 cos 72 + 1)
Most Upvoted Answer
In a regular pentagon PQRST, what is the ratio of the area of triangle...
To find the ratio of the area of triangle PRS to the area of pentagon PQRST, let's first calculate the areas of both the triangle and the pentagon.
Finding the area of triangle PRS:
The area of a triangle can be calculated using the formula: Area = (1/2) * base * height.
In triangle PRS, the base can be taken as PR and the height can be taken as the perpendicular distance from point S to line PR. Since the pentagon is regular, the perpendicular distance from the center of the pentagon to any of its sides is equal to the apothem (distance from the center to a side). Let's denote the apothem as 'a'.
We can divide triangle PRS into two right-angled triangles by drawing a perpendicular from point S to line PR. Let's call the point of intersection as X.
Now, the base PR can be divided into two parts: PX and RX. Each of these parts has a length equal to half the length of one side of the pentagon. Let's call the length of one side of the pentagon as 's'.
Since the pentagon is regular, the angle PSR is 72 degrees. Therefore, the angle PXR is 36 degrees (as it is half of the angle PSR).
Using trigonometry, we can find the length of PX and RX:
PX = (1/2) * s * cos(36)
RX = (1/2) * s * cos(36)
The height of triangle PRS (perpendicular distance from S to line PR) is equal to the apothem 'a'.
Now, we can calculate the area of triangle PRS:
Area of PRS = (1/2) * (PX + RX) * a
Area of PRS = (1/2) * (s * cos(36) + s * cos(36)) * a
Area of PRS = s * cos(36) * a
Finding the area of pentagon PQRST:
The area of a regular pentagon can be calculated using the formula: Area = (5/2) * s * a, where 's' is the length of one side of the pentagon and 'a' is the apothem.
Therefore, the area of pentagon PQRST = (5/2) * s * a
Ratio of the area of triangle PRS to the area of pentagon PQRST:
Ratio = (Area of PRS) / (Area of PQRST)
Ratio = (s * cos(36) * a) / ((5/2) * s * a)
Ratio = (2 * cos(36)) / 5
Simplifying the ratio:
Using the trigonometric identity cos(72) = 1 / (2 * cos(36)), we can rewrite the ratio as:
Ratio = (2 * cos(36)) / 5
Ratio = 1 / (5 * cos(36) / 2)
Ratio = 1 / (4 * cos(72) * 1)
Hence, the ratio of the area of triangle PRS to the area of pentagon PQRST is 1 / (4 * cos(72) * 1), which matches option 'C'.
Finding the area of triangle PRS:
The area of a triangle can be calculated using the formula: Area = (1/2) * base * height.
In triangle PRS, the base can be taken as PR and the height can be taken as the perpendicular distance from point S to line PR. Since the pentagon is regular, the perpendicular distance from the center of the pentagon to any of its sides is equal to the apothem (distance from the center to a side). Let's denote the apothem as 'a'.
We can divide triangle PRS into two right-angled triangles by drawing a perpendicular from point S to line PR. Let's call the point of intersection as X.
Now, the base PR can be divided into two parts: PX and RX. Each of these parts has a length equal to half the length of one side of the pentagon. Let's call the length of one side of the pentagon as 's'.
Since the pentagon is regular, the angle PSR is 72 degrees. Therefore, the angle PXR is 36 degrees (as it is half of the angle PSR).
Using trigonometry, we can find the length of PX and RX:
PX = (1/2) * s * cos(36)
RX = (1/2) * s * cos(36)
The height of triangle PRS (perpendicular distance from S to line PR) is equal to the apothem 'a'.
Now, we can calculate the area of triangle PRS:
Area of PRS = (1/2) * (PX + RX) * a
Area of PRS = (1/2) * (s * cos(36) + s * cos(36)) * a
Area of PRS = s * cos(36) * a
Finding the area of pentagon PQRST:
The area of a regular pentagon can be calculated using the formula: Area = (5/2) * s * a, where 's' is the length of one side of the pentagon and 'a' is the apothem.
Therefore, the area of pentagon PQRST = (5/2) * s * a
Ratio of the area of triangle PRS to the area of pentagon PQRST:
Ratio = (Area of PRS) / (Area of PQRST)
Ratio = (s * cos(36) * a) / ((5/2) * s * a)
Ratio = (2 * cos(36)) / 5
Simplifying the ratio:
Using the trigonometric identity cos(72) = 1 / (2 * cos(36)), we can rewrite the ratio as:
Ratio = (2 * cos(36)) / 5
Ratio = 1 / (5 * cos(36) / 2)
Ratio = 1 / (4 * cos(72) * 1)
Hence, the ratio of the area of triangle PRS to the area of pentagon PQRST is 1 / (4 * cos(72) * 1), which matches option 'C'.
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