Question

ii. solve the unknown in each triangle.

1.
in triangle abc, angle a = 35^{\circ}, angle b = 88^{\circ}, side ac = 44\text{ mm}, and side ab = x.

2.
in triangle abc, angle b = 51^{\circ}, angle a = 71^{\circ}, and side bc = 9.8\text{ cm}.

3.
in triangle xyz, angle x = 22^{\circ}, angle z = 39^{\circ}, and side xz = 29\text{ mm}.

4.
in triangle gma, angle g = 48^{\circ}, angle m = 61^{\circ}, side gm = 21\text{ cm}, and side ga = x.

5.
in triangle qrs, angle q = 98^{\circ}, side qs = 15\text{ m}, and side rs = 17.5\text{ m}.

6.
in triangle ghi, angle g = 115^{\circ}, side gh = 8\text{ cm}, and side hi = 13\text{ cm}.

Explanation:

🆕 New Concept Discovered: Law of Sines · Law of Cosines
*Using relationships between sides and angles in non-right triangles*

Here are the step-by-step solutions for each of the triangles shown in the image.

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Triangle 1: Solve for \( x \)

Step 1: Identify the given information

* We have a triangle \( \triangle ABC \).
* Angle \( A = 35^\circ \)
* Angle \( B = 88^\circ \)
* Side opposite to Angle \( B \) is \( b = 44\text{ mm} \)
* Side opposite to Angle \( C \) is \( c = x \)

Step 2: Find the third angle \( C \)

The sum of angles in a triangle is \( 180^\circ \):
\[ C = 180^\circ - (35^\circ + 88^\circ) = 180^\circ - 123^\circ = 57^\circ \]

Step 3: Apply the Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles:
\[ \frac{c}{\sin C} = \frac{b}{\sin B} \]

Substitute the known values:
\[ \frac{x}{\sin 57^\circ} = \frac{44}{\sin 88^\circ} \]

Solve for \( x \):
\[ x = \frac{44 \cdot \sin 57^\circ}{\sin 88^\circ} \]
\[ x \approx \frac{44 \cdot 0.8387}{0.9994} \approx 36.92\text{ mm} \]

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Triangle 2: Solve for the unknown side \( AC \) (let's call it \( b \))

Step 1: Identify the given information

* Angle \( B = 51^\circ \)
* Angle \( A = 71^\circ \)
* Side opposite to Angle \( A \) is \( a = 9.8\text{ cm} \) (side \( BC \))
* We need to find the side opposite to Angle \( B \), which is \( b \) (side \( AC \))

Step 2: Apply the Law of Sines

\[ \frac{b}{\sin B} = \frac{a}{\sin A} \]
\[ \frac{b}{\sin 51^\circ} = \frac{9.8}{\sin 71^\circ} \]

Step 3: Calculate \( b \)

\[ b = \frac{9.8 \cdot \sin 51^\circ}{\sin 71^\circ} \]
\[ b \approx \frac{9.8 \cdot 0.7771}{0.9455} \approx 8.05\text{ cm} \]

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Triangle 3: Solve for the unknown side \( YZ \) (let's call it \( x \))

Step 1: Identify the given information

* Angle \( X = 22^\circ \)
* Angle \( Z = 39^\circ \)
* Side opposite to Angle \( Y \) is \( y = 29\text{ mm} \) (side \( XZ \))
* We need to find the side opposite to Angle \( X \), which is \( x \) (side \( YZ \))

Step 2: Find the third angle \( Y \)

\[ Y = 180^\circ - (22^\circ + 39^\circ) = 180^\circ - 61^\circ = 119^\circ \]

Step 3: Apply the Law of Sines

\[ \frac{x}{\sin X} = \frac{y}{\sin Y} \]
\[ \frac{x}{\sin 22^\circ} = \frac{29}{\sin 119^\circ} \]

Step 4: Calculate \( x \)

\[ x = \frac{29 \cdot \sin 22^\circ}{\sin 119^\circ} \]
\[ x \approx \frac{29 \cdot 0.3746}{0.8746} \approx 12.42\text{ mm} \]

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Triangle 4: Solve for \( x \)

Step 1: Identify the given information

* Angle \( G = 48^\circ \)
* Angle \( M = 61^\circ \)
* Side opposite to Angle \( A \) is \( a = 21\text{ cm} \) (side \( GM \))
* We need to find the side opposite to Angle \( M \), which is \( x \) (side \( GA \))

Step 2: Find the third angle \( A \)

\[ A = 180^\circ - (48^\circ + 61^\circ) = 180^\circ - 109^\circ = 71^\circ \]

Step 3: Apply the Law of Sines

\[ \frac{x}{\sin M} = \frac{a}{\sin A} \]
\[ \frac{x}{\sin 61^\circ} = \frac{21}{\sin 71^\circ} \]

Step 4: Calculate \( x \)

\[ x = \frac{21 \cdot \sin 61^\circ}{\sin 71^\circ} \]
\[ x \approx \frac{21 \cdot 0.8746}{0.9455} \approx 19.43\text{ cm} \]

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Triangle 5: Solve for the unknown angle \( S \)

Step 1: Identify the given information

* Angle \( Q = 98^\circ \)
* Side opposite to Angle \( Q \) is \( q = 17.5\text{ m} \) (side \( RS \))
* Side opposite to Angle \( S \) is \( s = 15\text{ m} \) (side \( QR \))

Step 2: Apply the Law of Sines

\[ \frac{\sin S}{s} = \frac{\sin Q}{q} \]
\[ \frac{\sin S}{15} = \frac{\sin 98^\circ}{17.5} \]

Step 3: Calculate Angle \( S \)

\[ \sin S = \frac{15 \cdot \sin 98^\circ}{17.5} \]
\[ \sin S \approx \frac{15 \cdot 0.9903}{17.5} \approx 0.8488 \]
\[ S = \sin^{-1}(0.8488) \approx 58.08^\circ \]

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Triangle 6: Solve for the unknown angle \( I \)

Step 1: Identify the given information

* Angle \( G = 115^\circ \)
* Side opposite to Angle \( G \) is \( g = 13\text{ cm} \) (side \( HI \))
* Side opposite to Angle \( I \) is \( i = 8\text{ cm} \) (side \( GH \))

Step 2: Apply the Law of Sines

\[ \frac{\sin I}{i} = \frac{\sin G}{g} \]
\[ \frac{\sin I}{8} = \frac{\sin 115^\circ}{13} \]

Step 3: Calculate Angle \( I \)

\[ \sin I = \frac{8 \cdot \sin 115^\circ}{13} \]
\[ \sin I \approx \frac{8 \cdot 0.9063}{13} \approx 0.5577 \]
\[ I = \sin^{-1}(0.5577) \approx 33.90^\circ \]

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Answer:

1. Triangle 1: \( x \approx 36.92\text{ mm} \)
2. Triangle 2: \( AC \approx 8.05\text{ cm} \)
3. Triangle 3: \( YZ \approx 12.42\text{ mm} \)
4. Triangle 4: \( x \approx 19.43\text{ cm} \)
5. Triangle 5: \( \text{Angle } S \approx 58.08^\circ \)
6. Triangle 6: \( \text{Angle } I \approx 33.90^\circ \)