Question

ii. solve the unknown in each triangle.
1.
triangle abc with angle a = 35 degrees, angle b = 88 degrees, side ac = 44mm, and side ab = x.

2.
triangle abc with angle b = 51 degrees, angle a = 71 degrees, side bc = 9.8cm.

3.
triangle xyz with angle x = 22 degrees, angle z = 39 degrees, side xz = 29mm.

4.
triangle gma with angle g = 48 degrees, angle m = 61 degrees, side gm = 21cm, and side ga = x.

5.
triangle qrs with angle q = 98 degrees, side qs = 15m, side rs = 17.5m.

6.
triangle ghi with angle g = 115 degrees, side gh = 8cm, side hi = 13cm.

Explanation:

🆕 New Concept Discovered: Law of Sines
Using ratios of sides and opposite angles in non-right triangles.

Step 1: Identify the given information for Triangle 1

In triangle \(ABC\):

• Angle \(A = 35^\circ\)
• Angle \(B = 88^\circ\)
• Side opposite to Angle \(B\) is \(b = 44\text{ mm}\) (side \(AC\))
• Side opposite to Angle \(C\) is \(c = x\) (side \(AB\))

We need to find the value of \(x\).

Step 2: Find the third angle, Angle C

The sum of angles in any triangle is always \(180^\circ\). We can find Angle \(C\) by subtracting the known angles from \(180^\circ\):

$$ C = 180^\circ - (35^\circ + 88^\circ) $$

$$ C = 180^\circ - 123^\circ = 57^\circ $$

Step 3: Set up the Law of Sines

Since this is not a right-angled triangle, we cannot use basic SOH CAH TOA. Instead, we use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides:

$$ \frac{c}{\sin(C)} = \frac{b}{\sin(B)} $$

Substitute the known values into the formula:

$$ \frac{x}{\sin(57^\circ)} = \frac{44}{\sin(88^\circ)} $$

Step 4: Solve for x

Isolate \(x\) by multiplying both sides by \(\sin(57^\circ)\):

$$ x = \frac{44 \cdot \sin(57^\circ)}{\sin(88^\circ)} $$

Using a calculator to find the sine values:

• \(\sin(57^\circ) \approx 0.8387\)
• \(\sin(88^\circ) \approx 0.9994\)

Calculate the final value:

$$ x \approx \frac{44 \cdot 0.8387}{0.9994} \approx \frac{36.9028}{0.9994} \approx 36.92\text{ mm} $$

Answer:

$$ x \approx 36.9\text{ mm} $$