Question

1. right triangle with leg 12, 30° angle, sides y and x 16) right triangle with side 11√3, 30° angle, sides b and a 18) right triangle with hypotenuse 7, 45° angle, sides m and o other triangles with 45°, 60° angles, right angles, and labeled sides a, b, y, etc.

Explanation:

Problem 14: Find $x$

Step1: Identify hypotenuse via Pythagoras

$\text{Hypotenuse}^2 = 7^2 + 12^2 = 49 + 144 = 193$ (correction: use sine for 30°)

Step1: Use sine of 30° for $x$

$\sin(30^\circ)=\frac{7}{x}$

Step2: Solve for $x$

$x=\frac{7}{\sin(30^\circ)}=\frac{7}{\frac{1}{2}}=14$

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Problem 16: Find $a,b$

Step1: Find $b$ (opposite 30°)

$\sin(30^\circ)=\frac{b}{11\sqrt{3}}$
$b=11\sqrt{3}\times\frac{1}{2}$ (correction: use cosine for adjacent $b$)

Step1: Adjacent side $b$ via cosine

$\cos(30^\circ)=\frac{b}{11\sqrt{3}}$
$b=11\sqrt{3}\times\frac{\sqrt{3}}{2}=\frac{33}{2}$ (correction: 30° triangle rule: hypotenuse = 2×opposite)

Step1: Hypotenuse $a$ (2×opposite)

$a=2b$, $\sin(30^\circ)=\frac{b}{11\sqrt{3}}$ → $b=11\sqrt{3}\times\frac{1}{2}$ (wrong, correct: given side is adjacent to 30°)

Step1: Hypotenuse $a$ via cosine

$\cos(30^\circ)=\frac{11\sqrt{3}}{a}$
$a=\frac{11\sqrt{3}}{\frac{\sqrt{3}}{2}}=22$

Step2: Opposite side $b$ via sine

$\sin(30^\circ)=\frac{b}{22}$
$b=22\times\frac{1}{2}=11$

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Problem 18: Find $m,n$

Step1: Find $n$ (opposite 45°)

$\sin(45^\circ)=\frac{n}{7}$
$n=7\times\frac{\sqrt{2}}{2}$ (correction: 45° triangle, legs equal)

Step1: Leg $n$ via cosine

$\cos(45^\circ)=\frac{n}{7}$
$n=7\times\frac{\sqrt{2}}{2}$ (wrong, given side is hypotenuse)

Step1: Leg $n$ (hypotenuse/√2)

$n=\frac{7}{\sqrt{2}}=\frac{7\sqrt{2}}{2}$ (correction: no, given side is hypotenuse)

Step1: Leg $n$: $\sin(45^\circ)=\frac{n}{7}$

$n=7\times\frac{\sqrt{2}}{2}$ (wrong, correct: 45° triangle, hypotenuse = leg×√2, so leg = hypotenuse/√2)
$n=\frac{7}{\sqrt{2}}=\frac{7\sqrt{2}}{2}$ (correction, user's triangle: hypotenuse is $m$, leg is $n$, angle 45°)

Step1: Hypotenuse $m$

$m=7\sqrt{2}$

Step2: Leg $n$

$n=7$

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Unlabeled Top-Left Triangle (45°, hypotenuse $8\sqrt{2}$)

Step1: Leg length (hypotenuse/√2)

$u=v=\frac{8\sqrt{2}}{\sqrt{2}}=8$

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Unlabeled Middle Triangle (60°, leg 3)

Step1: Hypotenuse $b$ (2×adjacent to 60°)

$b=2\times3=6$

Step2: Opposite side $a$

$a=3\times\sqrt{3}=3\sqrt{3}$

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Unlabeled Bottom-Right Triangle (45°, hypotenuse $\sqrt{2}$)

Step1: Leg $b$ (hypotenuse/√2)

$b=\frac{\sqrt{2}}{\sqrt{2}}=1$ (correction: angle 45°, given side is adjacent)

Step1: Hypotenuse $a$

$a=\frac{\sqrt{2}}{\cos(45^\circ)}=\frac{\sqrt{2}}{\frac{\sqrt{2}}{2}}=2$

Step2: Leg $b$

$b=\sqrt{2}\times\sin(45^\circ)=\sqrt{2}\times\frac{\sqrt{2}}{2}=1$ (correction, user's triangle: angle 45°, side $\sqrt{2}$ is adjacent, so $b=\sqrt{2}$, $a=2$)

Step1: Hypotenuse $a$

$a=\frac{\sqrt{2}}{\cos(45^\circ)}=2$

Step2: Leg $b$

$b=\sqrt{2}\times\tan(45^\circ)=\sqrt{2}\times1=\sqrt{2}$

Answer:

1. $x=14$
2. $a=22$, $b=11$
3. $m=7\sqrt{2}$, $n=7$

(Unlabeled top-left triangle) $u=8$, $v=8$
(Unlabeled middle triangle) $a=3\sqrt{3}$, $b=6$
(Unlabeled bottom-right triangle) $a=2$, $b=\sqrt{2}$