ACT Math — Geometry Domain Test
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Question 1
A right triangle has legs of lengths 8 and 15, as shown. What is the length of the hypotenuse?
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By the Pythagorean theorem, \(8^2+15^2=64+225=289=17^2\).
Answer: A
Question 2
Two parallel lines are cut by a transversal. One acute angle measures \(52^\circ\), as shown. What is the measure of an obtuse angle formed?
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An acute angle and an adjacent obtuse angle are supplementary: \(180^\circ-52^\circ=128^\circ\).
Answer: G
Question 3
A circle has radius 6, as shown. What is its circumference?
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Use the circumference formula \(C=2\pi r\).
Substitute \(r=6\): \(C=2\pi(6)=12\pi\).
Answer: C
Question 4
A rectangle has length 12 and width 7. What is its area?
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Use the rectangle area formula \(A=lw\).
Substitute \(l=12\) and \(w=7\): \(A=12(7)=84\).
Answer: J
Question 5
The triangle shown has base 14 and perpendicular height 9. What is its area?
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Use the triangle area formula \(A=\frac12 bh\).
Substitute \(b=14\) and \(h=9\): \(A=\frac12(14)(9)=63\).
Answer: A
Question 6
Two similar triangles have corresponding side lengths 6 and 15. If another side of the smaller triangle is 8, what is the corresponding side of the larger triangle?
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The scale factor is \(15/6=2.5\).
The corresponding side is \(8(2.5)=20\).
Answer: G
Question 7
A rectangular prism has dimensions 3, 4, and 10. What is its volume?
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Use the rectangular prism volume formula \(V=lwh\).
Substitute the side lengths: \(V=3(4)(10)=120\).
Answer: C
Question 8
Points \(A(-2,1)\) and \(B(4,1)\) are shown. What is the length of \(\overline{AB}\)?
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The points have the same \(y\)-coordinate.
The horizontal distance is \(4-(-2)=6\).
Answer: J
Question 9
A sector of a circle has radius 8 and central angle \(90^\circ\). 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 \(\pi(8^2)=64\pi\).
The sector is one-fourth of the circle, so its area is \(\frac14(64\pi)=16\pi\).
Answer: A
Question 10
A triangle has side lengths 7, 10, and \(x\). Which value could \(x\) have?
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The third side must satisfy \(|10-7|<x<10+7\): \(3<x<17\).
Only 12 works.
Answer: G
Question 11
A central angle measuring \(100^\circ\) and an inscribed angle \(x^\circ\) intercept the same arc, as shown. What is \(x\)?
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An inscribed angle is half the central angle intercepting the same arc, so \(x=50\).
Answer: C
Question 12
In the right triangle shown, what is \(\cos \theta\)?
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Relative to \(\theta\), the adjacent side is 12 and the hypotenuse is 13: \(\cos\theta=12/13\).
Answer: J
Question 13
From point \(P\) outside a circle, \(PT\) is tangent at \(T\). The radius \(OT=9\) and \(OP=15\). What is \(PT\)?
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A radius to a tangent point is perpendicular to the tangent: \(OPT\) is right.
Thus \(PT=\sqrt{15^2-9^2}=12\).
Answer: A
Question 14
A regular polygon has each interior angle equal to \(150^\circ\). How many sides does the polygon have?
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For a regular \(n\)-gon, \(180-360/n=150\).
Thus \(360/n=30\) and \(n=12\).
Answer: G
Question 15
A circle has center \((2,-1)\) and passes through \((5,3)\). What is the equation of the circle?
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The radius is the distance from \((2,-1)\) to \((5,3)\): \(\sqrt{3^2+4^2}=5\).
Thus \((x-2)^2+(y+1)^2=25\).
Answer: C
Question 16
The ellipse shown is centered at the origin, has horizontal vertices at \((\pm6,0)\), and vertical co-vertices at \((0,\pm3)\). Which equation represents it?
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For a horizontal ellipse centered at the origin, \(x^2/a^2+y^2/b^2=1\).
Here \(a=6\) and \(b=3\).
Answer: J
Question 17
A hyperbola centered at the origin has vertices \((\pm4,0)\) and asymptotes \(y=\pm\frac34x\). Which equation represents the hyperbola?
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For \(x^2/a^2-y^2/b^2=1\), the vertices give \(a=4\).
The asymptote slope is \(b/a=3/4\), so \(b=3\).
Answer: A
Question 18
An angle measures \(\frac{7\pi}{12}\) radians. What is its measure in degrees?
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Multiply by \(180^\circ/\pi\): \(\frac{7\pi}{12}\cdot\frac{180^\circ}{\pi}=105^\circ\).
Answer: G
Question 19
A triangle has vertices \((-2,1)\), \((5,1)\), and \((-2,6)\), as shown. What is its area?
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Use the triangle area formula \(A=\frac12 bh\).
The horizontal leg has length 7 and the vertical leg has length 5, so \(A=\frac12(7)(5)=17.5\).
Answer: C
Question 20
In the right triangle shown, \(\tan\theta=\frac35\). If the side adjacent to \(\theta\) has length 10, what is the length of the side opposite \(\theta\)?
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Since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=3/5\), the scale factor from 5 to 10 is 2.
The opposite side is 6.
Answer: J