SAT Math Practice Test: Circles and Right Triangle Trigonometry
- Circles and Right Triangle Trigonometry
- 10 questions
- 15 minutes
These 10 questions cover circles and right triangle trigonometry from the SAT Geometry and Trigonometry domain. For arcs and sectors, find the fraction of the full circle first, then apply it to the circumference or the area.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
(x − 3)² + (y + 2)² = 49
Which of the following gives the center and radius of the circle with the equation shown?
Correct answer: B
In (x − h)² + (y − k)² = r², the center is (h, k) = (3, −2) and r = √49 = 7. The choice with (−3, 2) flips the signs, and 49 is r², not r.
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Question 2
x² + y² − 6x + 8y = 0
What is the radius of the circle with the equation shown?
Correct answer: A
Completing the square gives (x − 3)² + (y + 4)² = 9 + 16 = 25, so the radius is √25 = 5. The choice 25 is the square of the radius.
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Question 3
A circle has a radius of 9 centimeters. What is the length, in centimeters, of an arc of this circle with a central angle of 80°?
Correct answer: B
The arc is 80/360 = 2/9 of the circumference 18π, so its length is 2/9 × 18π = 4π. The choice 18π is the whole circumference.
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Question 4
A sector of a circle with radius 6 has a central angle of π/3 radians. What is the area of the sector?
Correct answer: C
The sector is (π/3) ÷ 2π = 1/6 of the circle, and 1/6 × 36π = 6π. The choice 12π uses r²θ without the factor 1/2, and 2π is the arc length.
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Question 5
In right triangle ABC, angle C is the right angle and sin A = 5/13. What is tan B?
Correct answer: D
sin A = 5/13 means BC = 5 and AB = 13, so AC = 12 by the Pythagorean theorem. For angle B, the opposite side is AC and the adjacent side is BC, so tan B = 12/5; the choice 5/12 is tan A.
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Question 6
If sin(x°) = cos(32°) and 0 < x < 90, what is the value of x?
Correct answer: D
The sine of an angle equals the cosine of its complement, so x = 90 − 32 = 58. The choice 32 assumes sine and cosine of the same angle are equal, which is true only for 45°.
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Question 7
The hypotenuse of a 30°-60°-90° triangle is 14 inches long. What is the length, in inches, of the side opposite the 60° angle?
Correct answer: C
In a 30°-60°-90° triangle the sides are in the ratio 1 : √3 : 2, so the short leg is 7 and the side opposite 60° is 7√3. The choice 7 is the side opposite the 30° angle.
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Question 8
What is the measure of an angle of 225° in radians?
Correct answer: C
Multiply by π/180: 225 × π/180 = 5π/4. The choice 3π/4 is 135°, and 7π/4 is 315°.
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Question 9
An inscribed angle of a circle intercepts an arc that measures 110°. What is the measure of the inscribed angle?
Correct answer: A
An inscribed angle is half of its intercepted arc, so it measures 110 ÷ 2 = 55°. The choice 110° is the measure of a central angle that intercepts the same arc.
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Question 10
Point P is 13 units from the center O of a circle with radius 5. Segment PT is tangent to the circle at point T. What is the length of PT?
Correct answer: D
A tangent is perpendicular to the radius at the point of tangency, so triangle OTP has a right angle at T and PT = √(13² − 5²) = √144 = 12. The choice 8 simply subtracts 5 from 13.
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