SAT Math - Geometry & Trigonometry Domain Test
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Question 1
Two parallel lines are cut by a transversal. One of the acute angles formed measures \(58^\circ\). What is the measure of any obtuse angle formed?
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An acute angle and an adjacent obtuse angle are supplementary.
Therefore, the obtuse angle measures \(180^\circ-58^\circ=122^\circ\).
Answer: C
Question 2
A triangle has angle measures \(47^\circ\), \(68^\circ\), and \(x^\circ\). What is the value of \(x\)?
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The angles of a triangle sum to \(180^\circ\): \(x=180-47-68=65\).
Answer: D
Question 3
A right triangle has legs of lengths 8 and 15. What is the length of the hypotenuse?
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By the Pythagorean theorem, \(c=\sqrt{8^2+15^2}=\sqrt{289}=17\).
Answer: 17
Question 4
A circle has radius 6. What is its circumference?
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The circumference is \(C=2\pi r=2\pi(6)=12\pi\).
Answer: A
Question 5
A rectangle has length 11 and width 7. What is its area, in square units?
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Use the rectangle area formula \(A=lw\).
Substitute \(l=11\) and \(w=7\): \(A=11(7)=77\).
Answer: A
Question 6
In a right triangle, \(\sin \theta=\frac{3}{5}\). Which ratio could represent \(\cos \theta\) for the same acute angle \(\theta\)?
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A \(3\)-\(4\)-\(5\) right triangle has \(\sin\theta=3/5\).
The adjacent-to-hypotenuse ratio is \(\cos\theta=4/5\).
Answer: C
Question 7
A rectangular prism has length 5, width 4, and height 9. What is its volume, in cubic units?
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The volume is \(V=lwh=5(4)(9)=180\).
Answer: 180
Question 8
A central angle of a circle measures \(90^\circ\). What fraction of the circle is the corresponding sector?
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A full circle is \(360^\circ\).
Thus the sector is \(90/360=1/4\) of the circle.
Answer: A
Question 9
Two similar triangles have corresponding side lengths 6 and 15. If another side of the smaller triangle has length 10, what is the length of the corresponding side of the larger triangle?
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The scale factor from the smaller triangle to the larger is \(15/6=5/2\).
Thus the corresponding side is \(10(5/2)=25\).
Answer: D
Question 10
A circle has center \((2,-3)\) and passes through \((8,-3)\). What is the radius of the circle?
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The two points have the same \(y\)-coordinate.
The radius is the horizontal distance \(|8-2|=6\).
Answer: 6
Question 11
A sector of a circle with radius 12 has a central angle of \(60^\circ\). What is the area of the sector?
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Use the circle area formula \(A=\pi r^2\).
The full circle area is \(\pi(12)^2=144\pi\).
The sector is \(60/360=1/6\) of the circle, so its area is \(\frac16(144\pi)=24\pi\).
Answer: B
Question 12
A 13-foot ladder rests against a vertical wall. The foot of the ladder is 5 feet from the wall. How many feet above the ground does the top of the ladder reach?
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If the other leg is \(x\), then \(x^2+5^2=13^2\).
Thus \(x^2=144\) and \(x=12\).
Answer: D
Question 13
An inscribed angle intercepts an arc measuring \(136^\circ\). What is the measure of the inscribed angle?
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An inscribed angle measures half its intercepted arc.
Its measure is \(136^\circ/2=68^\circ\).
Answer: B
Question 14
A \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle has a shorter leg of length 7. What is the length of its hypotenuse?
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In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the hypotenuse is twice the shorter leg.
Thus it is \(2(7)=14\).
Answer: 14
Question 15
A cylinder has radius 3 and height 8. What is its volume?
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The volume is \(V=\pi r^2h=\pi(3)^2(8)=72\pi\).
Answer: D
Question 16
In right triangle \(ABC\), angle \(C\) is \(90^\circ\), \(AC=12\), and \(BC=5\). What is \(\tan A\)?
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Relative to angle \(A\), the opposite side is \(BC=5\) and the adjacent leg is \(AC=12\).
Therefore, \(\tan A=5/12\).
Answer: B
Question 17
A circle in the \(xy\)-plane has equation \((x-4)^2+(y+1)^2=100\). A point on the circle has \(x\)-coordinate 10 and positive \(y\)-coordinate. What is the \(y\)-coordinate of the point?
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Substitute \(x=10\): \((10-4)^2+(y+1)^2=100\).
\(36+(y+1)^2=100\).
Then \((y+1)^2=64\), giving \(y+1=\pm8\). The positive \(y\)-coordinate is \(y=7\).
Answer: B
Question 18
A tangent segment from point \(P\) touches a circle at \(T\). The circle has center \(O\), \(OT=9\), and \(OP=15\). What is the length of \(PT\)?
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A radius to a point of tangency is perpendicular to the tangent: \(\triangle OTP\) is right.
Thus \(PT=\sqrt{15^2-9^2}=\sqrt{144}=12\).
Answer: C
Question 19
A sector has radius 10 and arc length \(5\pi\). What is the central angle of the sector, in degrees?
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Using \(s=r\theta\) with \(\theta\) in radians, \(5\pi=10\theta\): \(\theta=\pi/2\).
This equals \(90^\circ\).
Answer: 90
Question 20
A right triangle has acute angle \(\theta\) with \(\cos\theta=\frac{12}{13}\). A similar right triangle has hypotenuse 39. What is the length of the side opposite \(\theta\) in the larger triangle?
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Since \(\cos\theta=12/13\), the reference triangle is a \(5\)-\(12\)-\(13\) triangle.
The opposite side corresponds to 5.
The scale factor to hypotenuse 39 is \(39/13=3\), giving opposite side \(5(3)=15\).
Answer: C