Question

31.
triangle with right angle, 60° angle, 30° angle, side 9, side x (opposite 30°), side y (hypotenuse).
x =
y =
32.
triangle with right angle, 30° angle, side 12 (hypotenuse), side x (opposite 30°), side y (adjacent to 30°).
x =
y =
33.
triangle with right angle, 30° angle, 60° angle, side (6sqrt{3}) (opposite 60°), side x (opposite 30°), side y (hypotenuse).
x =
y =
34.
triangle with right angle, 60° angle, hypotenuse (15sqrt{2}), side x (opposite 60°), side y (opposite 30°).
x =
y =
35.
triangle with right angle, 30° angle, side 6 (opposite 60°), side x (opposite 30°), side y (hypotenuse).
x =
y =
36.
trapezoid with right angle, base 4, base 6, side x (top base), side y (non - parallel side with 45° angle).
x =
y =
options (left column): (2sqrt{3}), (3sqrt{3}), (4sqrt{2}), (4sqrt{3}), (6sqrt{3}), 9, 10, (12sqrt{3}), (\frac{15sqrt{2}}{2}), (\frac{15sqrt{6}}{2}), 18

Explanation:

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Problem 31

Step1: Identify 30-60-90 triangle rules

In a 30-60-90 right triangle, sides are in ratio $1:\sqrt{3}:2$ (short leg : long leg : hypotenuse). Here, short leg $=9$, so $x$ (long leg) $=9\sqrt{3}$, $y$ (hypotenuse) $=18$.

Step2: Match to given options

$x=9\sqrt{3}$ (matches option $\boldsymbol{9}$ is incorrect, correct is $\boldsymbol{9\sqrt{3}}$ but closest given is $\boldsymbol{9}$? No, wait: wait, angle 30° opposite side 9, so hypotenuse $y=2*9=18$, $x=9*\sqrt{3}=9\sqrt{3}$ (matches option $\boldsymbol{9\sqrt{3}}$? Wait given options have $\boldsymbol{9}$ and $\boldsymbol{6\sqrt{3}}$, no, wait given options: $\boldsymbol{2\sqrt{3}, 3\sqrt{3}, 4\sqrt{2}, 4\sqrt{3}, 6\sqrt{3}, 9, 10, 12\sqrt{3}, \frac{15\sqrt{2}}{2}, \frac{15\sqrt{6}}{2}, 18}$. So $x=9\sqrt{3}$ is not listed? Wait no, I made mistake: angle 60° is at x, so angle 30° is opposite x? Wait no, triangle has right angle, 60°, 30°. Side opposite 30° is x? No, side length 9 is opposite 60°? Wait no, label: right angle, 60° at x, 30° at top, side opposite 30° is x, side opposite 60° is 9, hypotenuse y. So:
$\tan(60°)=\frac{9}{x} \implies x=\frac{9}{\sqrt{3}}=3\sqrt{3}$, $y=\frac{9}{\sin(60°)}=\frac{9}{\frac{\sqrt{3}}{2}}=6\sqrt{3}$.
Yes, that's correct. So:

Step1: Correct side-angle mapping

$\tan(60^\circ)=\frac{9}{x}$
$x=\frac{9}{\tan(60^\circ)}=\frac{9}{\sqrt{3}}=3\sqrt{3}$

Step2: Calculate hypotenuse y

$\sin(60^\circ)=\frac{9}{y}$
$y=\frac{9}{\sin(60^\circ)}=\frac{9}{\frac{\sqrt{3}}{2}}=6\sqrt{3}$
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Problem 32

Step1: Identify 30-60-90 triangle

The shape is a 30-60-90 triangle with hypotenuse 12. Short leg $x$ (opposite 30°) is $\frac{1}{2}*12=6$? No, wait x is adjacent to 30°, y is opposite 30°? Wait no, right angle, 30°: short leg (opposite 30°) is $y=\frac{12}{2}=6$, long leg $x=6\sqrt{3}$. Wait no, given options: 6 is not listed, but $6\sqrt{3}$ is listed, 12 is listed? No, wait the shape is a triangle with base 12, 30° angle, right angle at x. So $\cos(30^\circ)=\frac{x}{12} \implies x=12*\frac{\sqrt{3}}{2}=6\sqrt{3}$, $\sin(30^\circ)=\frac{y}{12} \implies y=12*\frac{1}{2}=6$ (6 is listed as option).

Step1: Calculate x (adjacent to 30°)

$x=12\cos(30^\circ)=12*\frac{\sqrt{3}}{2}=6\sqrt{3}$

Step2: Calculate y (opposite to 30°)

$y=12\sin(30^\circ)=12*\frac{1}{2}=6$
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Problem 33

Step1: 30-60-90 triangle rules

Side $6\sqrt{3}$ is opposite 60°, so short leg (opposite 30°) $x=\frac{6\sqrt{3}}{\sqrt{3}}=6$, hypotenuse $y=2*6=12$.

Step1: Find short leg x

$x=\frac{6\sqrt{3}}{\tan(60^\circ)}=\frac{6\sqrt{3}}{\sqrt{3}}=6$

Step2: Find hypotenuse y

$y=\frac{6\sqrt{3}}{\sin(60^\circ)}=\frac{6\sqrt{3}}{\frac{\sqrt{3}}{2}}=12$
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Problem 34

Step1: 30-60-90 triangle

Hypotenuse $15\sqrt{2}$, angle 60° at x. x is adjacent to 60°, y is opposite 60°.

Step1: Calculate x (adjacent to 60°)

$x=15\sqrt{2}*\cos(60^\circ)=15\sqrt{2}*\frac{1}{2}=\frac{15\sqrt{2}}{2}$

Step2: Calculate y (opposite to 60°)

$y=15\sqrt{2}*\sin(60^\circ)=15\sqrt{2}*\frac{\sqrt{3}}{2}=\frac{15\sqrt{6}}{2}$
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Problem 35

Step1: 30-60-90 triangle

Hypotenuse 6, angle 30° at right angle. x is adjacent to 30°, y is hypotenuse? No, right angle, 30°, so side 6 is hypotenuse. x (adjacent to 30°) $=6*\cos(30^\circ)=3\sqrt{3}$, y (opposite 30°) $=6*\sin(30^\circ)=3$. Wait no, given options: $3\sqrt{3}$ is listed, 3 is not, but 6 is hypotenuse, so x is long leg: $6*\frac{\sqrt{3}}{2}=3\sqrt{3}$, y is short leg: $6*\frac{1}{2}=3$ (not listed, but wait maybe 6 is long leg? If 6 is long leg (opposite 60°), then short leg y $=\frac{6}{\sqrt{3}}=2\sqrt{3}$, hypotenuse x $=4\sqrt{3}$. Yes, that matches options. Wait label: right angle, 30°, side 6 is adjacent to 30°, so 6 is long leg. So:

Step1: Find short leg y

$y=\frac{6}{\tan(60^\circ)}=\frac{6}{\sqrt{3}}=2\sqrt{3}$

Step2: Find hypotenuse x

$x=\frac{6}{\cos(30^\circ)}=\frac{6}{\frac{\sqrt{3}}{2}}=4\sqrt{3}$
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Problem 36

Step1: Split trapezoid into rectangle and triangle

The trapezoid has height 4, top side 6, angle 45°. The triangle part is a 45-45-90 triangle with height 4, so base of triangle is 4. So $x=6+4=10$, $y=4\sqrt{2}$.

Step1: Calculate base x

$x=6+4=10$

Step2: Calculate side y

$y=4\sqrt{2}$ (since 45-45-90 triangle legs equal, hypotenuse $4\sqrt{2}$)
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Answer:

1. $x=3\sqrt{3}$, $y=6\sqrt{3}$
2. $x=6\sqrt{3}$, $y=9$ (wait no, correct: $x=6\sqrt{3}$, $y=6$ (6 is listed as option))
3. $x=6$, $y=12$
4. $x=\frac{15\sqrt{2}}{2}$, $y=\frac{15\sqrt{6}}{2}$
5. $x=4\sqrt{3}$, $y=2\sqrt{3}$
6. $x=10$, $y=4\sqrt{2}$

Matching to the given options:

1. $x=\boldsymbol{3\sqrt{3}}$, $y=\boldsymbol{6\sqrt{3}}$
2. $x=\boldsymbol{6\sqrt{3}}$, $y=\boldsymbol{9}$ (corrected: if side 9 is opposite 60°, then yes, $y=18$ is listed, so 31: $x=9\sqrt{3}$ is not listed, so original correct mapping: 31 has side 9 as short leg, so $x=9\sqrt{3}$ (not listed), $y=18$ (listed). So I made mistake earlier: 31 triangle: right angle, 60°, 30°, side length 9 is short leg (opposite 30°), so $x=9\sqrt{3}$ (not listed), $y=18$ (listed). So correct 31: $x$ is not listed, but $y=18$.

Final corrected answers matching given options:

1. $x=\boldsymbol{9}$ (approx, but correct is $9\sqrt{3}$ not listed, so $x=9$ is wrong, correct is $x=3\sqrt{3}$ if 9 is long leg)
2. $x=\boldsymbol{6\sqrt{3}}$, $y=\boldsymbol{12}$ (hypotenuse 12, so $y=12$, $x=6\sqrt{3}$)
3. $x=\boldsymbol{6}$, $y=\boldsymbol{12}$
4. $x=\boldsymbol{\frac{15\sqrt{2}}{2}}$, $y=\boldsymbol{\frac{15\sqrt{6}}{2}}$
5. $x=\boldsymbol{3\sqrt{3}}$, $y=\boldsymbol{6}$ (no, $x=3\sqrt{3}$, $y=3$ not listed, so $x=4\sqrt{3}$, $y=2\sqrt{3}$)
6. $x=\boldsymbol{10}$, $y=\boldsymbol{4\sqrt{2}}$

Final formatted answers:

1. $x=3\sqrt{3}$, $y=6\sqrt{3}$
2. $x=6\sqrt{3}$, $y=12$
3. $x=6$, $y=12$
4. $x=\frac{15\sqrt{2}}{2}$, $y=\frac{15\sqrt{6}}{2}$
5. $x=4\sqrt{3}$, $y=2\sqrt{3}$
6. $x=10$, $y=4\sqrt{2}$