Geometry & Trigonometry Domain Test 1

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Question 1

Two lines intersect at a point. One of the angles formed measures $47°$. What is the measure of the angle that is vertical to it?

A) $43°$
B) $47°$
C) $133°$
D) $143°$

Show solution

Vertical angles are formed opposite each other when two lines intersect. They are always equal in measure.

The vertical angle also measures $47°$.

1. $43°$: subtracted from 90 — this gives the complement, not the vertical angle
1. $133°$: subtracted from 180 — this gives the supplement, not the vertical angle
1. $143°$: subtracted from 190 — no geometric relationship

Answer: B

Question 2

A right triangle has legs of length $8$ and $15$. What is the length of the hypotenuse?

A) $\sqrt{23}$
B) $17$
C) $23$
D) $\sqrt{289}$

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\[c = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17\]

This is the well-known 8–15–17 Pythagorean triple.

Note: Choice D ($\sqrt{289}$) equals 17 but is an unsimplified form — choice B is the correct simplified answer.

1. $\sqrt{23}$: computed $\sqrt{8 + 15}$ instead of $\sqrt{8^2 + 15^2}$
1. $23$: added the legs without squaring: $8 + 15 = 23$
1. $\sqrt{289}$: correctly equals 17 but left unsimplified

Answer: B

Question 3

What is the area of a circle with radius $6$?

A) $6\pi$
B) $12\pi$
C) $36\pi$
D) $72\pi$

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\[A = \pi r^2 = \pi (6)^2 = 36\pi\]

1. $6\pi$: used $A = \pi r$ instead of $\pi r^2$
1. $12\pi$: used the circumference formula $C = 2\pi r = 12\pi$ instead of the area formula
1. $72\pi$: used $A = 2\pi r^2 = 72\pi$ (doubled the formula)

Answer: C

Question 4

A triangle has two angles measuring $52°$ and $73°$. What is the measure, in degrees, of the third angle?

Enter your answer:

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The sum of interior angles in any triangle is $180°$.

\[\text{Third angle} = 180 - 52 - 73 = 55°\]

Answer: 55

Question 5

In the figure below, line $m$ is parallel to line $n$, and line $t$ is a transversal. If angle 1 measures $112°$, what is the measure of angle 2?

Note: Figure not drawn to scale.

A) $68°$
B) $78°$
C) $112°$
D) $124°$

Show solution

Angles 1 and 2 are corresponding angles — they occupy the same position at each intersection (both are upper-left of the transversal). When two parallel lines are cut by a transversal, corresponding angles are equal.

\[\angle 2 = \angle 1 = 112°\]

1. $68°$: computed the supplement’s complement ($180 - 112 = 68$) — this would be the co-interior angle, not the corresponding angle
1. $78°$: no standard geometric relationship yields this value
1. $124°$: added $12°$ to $112°$ for no geometric reason

Answer: C

Question 6

What is the volume of a rectangular prism with length $8$, width $5$, and height $3$?

A) $40$
B) $79$
C) $120$
D) $158$

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\[V = l \times w \times h = 8 \times 5 \times 3 = 120\]

1. $40$: multiplied only two of the three dimensions ($8 \times 5 = 40$)
1. $79$: used surface area formula partially: $2(8\cdot5 + 5\cdot3 + 8\cdot3)/2$… or added $8 + 5 + 3 = 16$ and multiplied incorrectly
1. $158$: computed the full surface area: $2(40 + 15 + 24) = 2(79) = 158$

Answer: C

Question 7

In right triangle $PQR$, the right angle is at $Q$. If $\sin(\angle P) = \dfrac{5}{13}$, what is $\cos(\angle P)$?

A) $\dfrac{5}{12}$
B) $\dfrac{12}{13}$
C) $\dfrac{13}{12}$
D) $\dfrac{5}{13}$

Show solution

$\sin(\angle P) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{5}{13}$, so the opposite side $= 5$ and hypotenuse $= 13$.

By the Pythagorean theorem, the adjacent side $= \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.

\[\cos(\angle P) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}\]

1. $\dfrac{5}{12}$: divided opposite by adjacent — this is $\tan(\angle P)$, not $\cos(\angle P)$
1. $\dfrac{13}{12}$: inverted the cosine ratio (hypotenuse over adjacent)
1. $\dfrac{5}{13}$: restated $\sin(\angle P)$ instead of finding $\cos(\angle P)$

Answer: B

Question 8

A circle has a circumference of $18\pi$. What is the area of the circle?

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From the circumference: $C = 2\pi r = 18\pi \Rightarrow r = 9$.

Area $= \pi r^2 = \pi(9)^2 = 81\pi$.

Since the grid accepts $81\pi$, enter $81$ (the coefficient of $\pi$).

Answer: 81 (i.e., $81\pi$)

Question 9

In the figure below, triangle $ABC$ is similar to triangle $DEF$ with $AB$ corresponding to $DE$. If $AB = 4$, $AC = 6$, $BC = 5$, and $DE = 10$, what is the length of $EF$?

Note: Figure not drawn to scale.

A) $8$
B) $10$
C) $12.5$
D) $15$

Show solution

The scale factor from $\triangle ABC$ to $\triangle DEF$ is:

\[k = \frac{DE}{AB} = \frac{10}{4} = 2.5\]

$EF$ corresponds to $BC$, so:

\[EF = BC \times k = 5 \times 2.5 = 12.5\]

1. $8$: applied scale factor to $AC$ instead of $BC$, or used $k = \frac{4}{5}$ (inverted)
1. $10$: used the value of $DE$ (which is the corresponding side to $AB$, not $BC$)
1. $15$: applied scale factor to $AC$: $6 \times 2.5 = 15$ (this would be $DF$, not $EF$)

Answer: C

Question 10

A central angle of a circle measures $120°$ and the circle has a radius of $9$. What is the length of the arc subtended by this angle?

A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $27\pi$

Show solution

Arc length $= \dfrac{\theta}{360°} \times 2\pi r = \dfrac{120}{360} \times 2\pi(9) = \dfrac{1}{3} \times 18\pi = 6\pi$

1. $3\pi$: used $\frac{120}{360} \times \pi r = \frac{1}{3} \times 9\pi = 3\pi$ (forgot the factor of 2 in circumference)
1. $9\pi$: used $\frac{1}{3} \times 27\pi$ — computed the area of the sector instead of arc length, or used $r^2$ instead of $r$
1. $27\pi$: used the full circumference $18\pi$ but wrote $27\pi$… or applied no fraction at all: $3 \times 9\pi = 27\pi$

Answer: B

Question 11

A regular hexagon has interior angles that all have the same measure. What is the measure of each interior angle of a regular hexagon?

A) $108°$
B) $120°$
C) $135°$
D) $144°$

Show solution

Sum of interior angles of a polygon with $n$ sides $= (n-2) \times 180°$.

For a hexagon ($n = 6$): $(6-2) \times 180 = 4 \times 180 = 720°$.

Each interior angle of a regular hexagon $= \dfrac{720}{6} = 120°$.

1. $108°$: this is the interior angle of a regular pentagon ($n = 5$)
1. $135°$: this is the interior angle of a regular octagon ($n = 8$)
1. $144°$: this is the interior angle of a regular decagon ($n = 10$)

Answer: B

Question 12

In the figure below, a square with side length $10$ has a circle inscribed in it (the circle is tangent to all four sides). What is the area of the shaded region between the square and the circle?

The shaded region is the area of the square minus the area of the circle.

A) $100 - 5\pi$
B) $100 - 10\pi$
C) $100 - 25\pi$
D) $100 - 100\pi$

Show solution

The circle is inscribed in the square of side $10$, so the circle’s diameter equals the side length: diameter $= 10 \Rightarrow r = 5$.

Area of square $= 10^2 = 100$.

Area of circle $= \pi r^2 = 25\pi$.

Shaded area $= 100 - 25\pi$.

1. $100 - 5\pi$: used $r = 5$ but computed $\pi r = 5\pi$ instead of $\pi r^2$
1. $100 - 10\pi$: used the diameter ($10$) as the radius: $(10)^2/= $ no… used $\pi \cdot 10 = 10\pi$
1. $100 - 100\pi$: used the side length ($10$) as the radius: $\pi(10)^2 = 100\pi$

Answer: C

Question 13

In a right triangle, one acute angle measures $30°$ and the hypotenuse has length $20$. What is the length of the side opposite the $30°$ angle?

Enter your answer:

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In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$, where the hypotenuse corresponds to $2$.

If hypotenuse $= 20$, the scale factor is $\frac{20}{2} = 10$.

Side opposite $30°$ (shortest leg) $= 1 \times 10 = 10$.

Alternatively: $\sin(30°) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{1}{2}$, so opposite $= \dfrac{1}{2} \times 20 = 10$.

Answer: 10

Question 14

An inscribed angle in a circle intercepts an arc of $140°$. What is the measure of the inscribed angle?

A) $35°$
B) $70°$
C) $140°$
D) $280°$

Show solution

The Inscribed Angle Theorem states that an inscribed angle is half the measure of its intercepted arc.

\[\text{Inscribed angle} = \frac{140°}{2} = 70°\]

1. $35°$: halved twice ($140 \div 4 = 35$)
1. $140°$: confused inscribed angle with central angle — a central angle equals the arc, but an inscribed angle is half
1. $280°$: doubled the arc measure instead of halving

Answer: B

Question 15

In right triangle $ABC$, the right angle is at $C$, $AB = 13$, and $BC = 5$. What is $\tan(\angle A)$?

Note: Figure not drawn to scale.

A) $\dfrac{5}{13}$
B) $\dfrac{12}{13}$
C) $\dfrac{5}{12}$
D) $\dfrac{12}{5}$

Show solution

$AC = \sqrt{AB^2 - BC^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.

For $\angle A$: the opposite side is $BC = 5$ and the adjacent side is $AC = 12$.

\[\tan(\angle A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{5}{12}\]

1. $\dfrac{5}{13}$: this is $\sin(\angle A)$ (opposite over hypotenuse)
1. $\dfrac{12}{13}$: this is $\cos(\angle A)$ (adjacent over hypotenuse)
1. $\dfrac{12}{5}$: inverted the tangent ratio (adjacent over opposite) — this equals $\tan(\angle B)$

Answer: C

Question 16

The equation of a circle in the $xy$-plane is $(x - 3)^2 + (y + 1)^2 = 25$. A point $P$ lies on the circle. Which of the following could be the coordinates of $P$?

A) $(3, 5)$
B) $(8, -1)$
C) $(3, 6)$
D) $(0, 0)$

Show solution

The circle has center $(3, -1)$ and radius $5$. Substitute each point into $(x-3)^2 + (y+1)^2$ and check whether it equals $25$:

1. $(3, 5)$: $(0)^2 + (6)^2 = 36 \neq 25$
1. $(8, -1)$: $(5)^2 + (0)^2 = 25$ ✓
1. $(3, 6)$: $(0)^2 + (7)^2 = 49 \neq 25$
1. $(0, 0)$: $(-3)^2 + (1)^2 = 9 + 1 = 10 \neq 25$

Only $(8, -1)$ lies on the circle.

Answer: B

Question 17

An angle measures $\dfrac{5\pi}{6}$ radians. What is its measure in degrees?

Enter your answer:

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To convert radians to degrees, multiply by $\dfrac{180°}{\pi}$:

\[\frac{5\pi}{6} \times \frac{180°}{\pi} = \frac{5 \times 180°}{6} = \frac{900°}{6} = 150°\]

Answer: 150

Question 18

In right triangle $XYZ$, the right angle is at $Y$, and $\angle X = 40°$. Which of the following is equal to $\cos(40°)$?

A) $\sin(40°)$
B) $\cos(50°)$
C) $\sin(50°)$
D) $\tan(40°)$

Show solution

In a right triangle, the two acute angles are complementary (they sum to $90°$). The complementary angle identity states:

\[\cos(\theta) = \sin(90° - \theta)\]

\[\cos(40°) = \sin(90° - 40°) = \sin(50°)\]

1. $\sin(40°)$: these are not equal unless $\theta = 45°$; $\cos(40°) \approx 0.766$ while $\sin(40°) \approx 0.643$
1. $\cos(50°)$: $\cos(50°) \approx 0.643 \neq \cos(40°) \approx 0.766$; this equals $\sin(40°)$, not $\cos(40°)$
1. $\tan(40°) \approx 0.839 \neq \cos(40°)$

Answer: C

Question 19

In the figure below, parallel lines $m$ and $n$ are cut by transversal $t$. The measure of angle 1 is $(2x + 40)°$ and the measure of angle 2 is $(4x + 20)°$.

Note: Figure not drawn to scale.

What is the measure of angle 2?

A) $80°$
B) $90°$
C) $100°$
D) $110°$

Show solution

Angles 1 and 2 are co-interior (same-side interior) angles — both between the parallel lines on the same side of the transversal. Co-interior angles are supplementary:

\[(2x + 40) + (4x + 20) = 180\] \[6x + 60 = 180\] \[6x = 120 \Rightarrow x = 20\]

\[\angle 2 = 4(20) + 20 = 80 + 20 = 100°\]

Verify: $\angle 1 = 2(20) + 40 = 80°$, and $80 + 100 = 180°$ ✓

1. $80°$: found the value of $\angle 1$ instead of $\angle 2$
1. $90°$: arithmetic error — solved $6x = 120$ as $x = 15$ giving $\angle 2 = 80°$, or other slip
1. $110°$: solved correctly for $x$ but used $\angle 2 = 4(20) + 30 = 110$ (wrong constant)

Answer: C

Question 20

A cylinder has a radius of $4$ and a height of $9$. What is the volume of the cylinder? (Enter the numerical coefficient of $\pi$.)

Enter your answer:

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\[V = \pi r^2 h = \pi (4)^2 (9) = \pi \times 16 \times 9 = 144\pi\]

The coefficient of $\pi$ is $144$.

Answer: 144