Question 1 :
Find the length of the arc that is bolded. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 2 :
In the diagram given below, if QRS is a central angle and m∠QRS = 81°, m∠SRT = 115°, and radius is 5 cm, then find the length of the arc QST. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 3 :
If m∠LMN = 19° and radius is 15 inches, then find the length of arc LN. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 4 :
Find the length of the arc highlighted in red color. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 5 :
Find the area of the sector that is outlined with the bold line. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 6 :
In circle C, if XCZ is a central angle and XYZ is an inscribed angle and m∠XYZ = 58° and radius is 10 inches. Find the area of sector XCZ. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 7 :
If QRS is a central angle and m∠QRS = 46°, m∠SRT = 80°, and diameter is 4 inches, then find the area of the shaded sector. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Question 1 :
Find the length of the arc that is bolded. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2πr
Substitute r = 8, arc Measure = 315° and π ≈ 3.14.
≈ (315° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8
= 44
So, the length of the arc is about 44 cm.
Question 2 :
In the diagram given below, if QRS is a central angle and m∠QRS = 81°, m∠SRT = 115°, and radius is 5 cm, then find the length of the arc QST. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
To find the length of the arc QST, first we have to find the arc measure QST or the central angle m∠QRT.
m∠QRT = m∠QRS + m∠SRT
m∠QRT = 81° + 115°
m∠QRT = 196°
The formula to find the arc length is
= (Central Angle / 360°) ⋅ 2πr
Substitute r = 5, Central Angle = 196° and π ≈ 3.14.
≈ (196° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5
= 17.1
So, the length of the arc is about 17.1 cm.
Question 3 :
If m∠LMN = 19° and radius is 15 inches, then find the length of arc LN. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
To find the length of the arc LN, first we have to find the arc measure LN.
By Inscribed Angle Theorem, we have
1/2 ⋅ Arc Measure = m∠LMN
Multiply both sides by 2.
Arc Measure = 2 ⋅ m∠LMN
Arc Measure = 2 ⋅ 19°
Arc Measure = 38°
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2πr
Substitute r = 15, Arc Measure = 38° and π ≈ 3.14.
≈ (38° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 15
= 9.9
So, the length of the arc is about 9.9 inches.
Question 4 :
Find the length of the arc highlighted in red color. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
From the given diagram, we have
m∠MCN + Measure of arc MON = 360°
Plug m∠MCN = 88°
88° + Measure of arc MON = 360°
Subtract 88° from both sides.
Measure of arc MON = 272°
Given : Diameter is 4 inches.
Then, the radius is
= Diameter / 2
= 10 / 2
= 5 ft
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2πr
Substitute r = 5, Arc Measure = 272° and Π ≈ 3.14.
≈ (272° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5
= 23.7 ft
So, the length of the arc is about 23.7 ft.
Question 5 :
Find the area of the sector that is outlined with the bold line. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
The formula to find area of the sector is
= (θ / 360°) ⋅ πr2
Substitute r = 11, θ = 300° and π ≈ 3.14.
≈ (300° / 360°) ⋅ 3.14 ⋅ 112
= 316.7
So, the area of the given sector is about 316.7 cm².
Question 6 :
In circle C, if XCZ is a central angle and XYZ is an inscribed angle and m∠XYZ = 58° and radius is 10 inches. Find the area of sector XCZ. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
By Inscribed Angle Theorem, we have
1/2 ⋅ m∠XCZ = m∠XYZ
Multiply both sides by 2.
m∠XCZ = 2 ⋅ m∠XYZ
Given : m∠XYZ = 58°.
Then, we have
m∠XCZ = 2 ⋅ 58°
m∠XCZ = 116°
So, the central angle θ is 116°.
The formula to find area of the sector is
= (θ / 360°) ⋅ πr2
Substitute r = 10, θ = 116° and Π ≈ 3.14.
≈ (116° / 360°) ⋅ 3.14 ⋅ 102
= 101.2
So, the area of sector XCZ is about 101.2 in2.
Question 7 :
If QRS is a central angle and m∠QRS = 46°, m∠SRT = 80°, and diameter is 4 inches, then find the area of the shaded sector. (Take π ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
Given : m∠QRS = 46° and m∠SRT = 80°.
Then, we have
m∠QRS + m∠SRT = 46° + 80°
m∠QRS + m∠SRT = 126°
Measure of central angle of the shaded region :
m∠QRT = 360° - 126°
m∠QRT = 234°
Radius of the circle :
Radius = Diameter / 2
Radius = 4 / 2
Radius = 2 inches
The formula to find area of the sector is
= (θ / 360°) ⋅ πr2
Substitute r = 2, θ = 234° and Π ≈ 3.14
≈ (234° / 360°) ⋅ 3.14 ⋅ 22
= 8.2
So, the area of the shaded sector is about 8.2 in2.
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