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Solve the following triangles and find their areas.

1. Triangle ABC in which ∠A = 29°, ∠B = 68° and b = 27mm.

2. Triangle ABC in which ∠B = 71°26', ∠C = 56°22', and b = 8.60cm.

3. Triangle DEF in which d = 17cm, f = 22cm and F = 26°.

4. Triangle DEF in which d = 32.6mm, e = 25.4mm, and ∠D = 104°22'.

Answer :

To solve these triangle problems and find their areas, we will use the Law of Sines and formulas for triangle area. Let's go through each triangle one by one.

1. Triangle ABC:

  • Given: [tex]\angle A = 29^\circ[/tex], [tex]\angle B = 68^\circ[/tex], [tex]b = 27 \text{ mm}[/tex].

  • First, find [tex]\angle C[/tex] using the angle sum property of triangles:
    [tex]\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 29^\circ - 68^\circ = 83^\circ.[/tex]

  • Using the Law of Sines:
    [tex]\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.[/tex]

  • Find [tex]c[/tex]:
    [tex]c = \frac{b \cdot \sin C}{\sin B} = \frac{27 \cdot \sin 83^\circ}{\sin 68^\circ}.[/tex]

  • Find [tex]a[/tex]:
    [tex]a = \frac{b \cdot \sin A}{\sin B} = \frac{27 \cdot \sin 29^\circ}{\sin 68^\circ}.[/tex]

  • Calculate the area using Heron's formula:
    [tex]s = \frac{a + b + c}{2},[/tex]
    [tex]\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}.[/tex]

2. Triangle ABC:

  • Given: [tex]\angle B = 71^\circ 26'[/tex], [tex]\angle C = 56^\circ 22'[/tex], [tex]b = 8.60 \text{ cm}[/tex].

  • Convert minutes into decimal degrees:
    [tex]71^\circ 26' = 71.433^\circ, \; 56^\circ 22' = 56.367^\circ.[/tex]

  • Find [tex]\angle A[/tex]:
    [tex]\angle A = 180^\circ - 71.433^\circ - 56.367^\circ = 52.2^\circ.[/tex]

  • Use the Law of Sines to find sides [tex]a[/tex] and [tex]c[/tex]:
    [tex]a = \frac{b \cdot \sin A}{\sin B}, \; c = \frac{b \cdot \sin C}{\sin B}.[/tex]

  • Calculate the area with Heron's formula.

3. Triangle DEF:

  • Given: [tex]d = 17 \text{ cm}[/tex], [tex]f = 22 \text{ cm}[/tex], [tex]F = 26^\circ[/tex].

  • Use the Law of Cosines to find [tex]e[/tex]:
    [tex]e^2 = d^2 + f^2 - 2df \cdot \cos F.[/tex]

  • Use the Law of Sines to find [tex]\angle D[/tex] and [tex]\angle E[/tex].

  • Area of triangle:
    [tex]\text{Area} = \frac{1}{2} \cdot d \cdot f \cdot \sin F.[/tex]

4. Triangle DEF:

  • Given: [tex]d = 32.6 \text{ mm}[/tex], [tex]e = 25.4 \text{ mm}[/tex], [tex]\angle D = 104^\circ 22'[/tex].

  • Convert minutes into decimal degrees:
    [tex]104^\circ 22' = 104.367^\circ.[/tex]

  • Use the Law of Cosines to find [tex]f[/tex]:
    [tex]f^2 = d^2 + e^2 - 2de \cdot \cos D.[/tex]

  • Use the Law of Sines to find [tex]\angle E[/tex] and [tex]\angle F[/tex].

  • Calculate the area using:
    [tex]\text{Area} = \frac{1}{2} \cdot d \cdot e \cdot \sin D.[/tex]

These steps provide an approach to solve each triangle and calculate their areas. Remember to calculate each value using a calculator to obtain numerical results for lengths and areas.

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Rewritten by : Barada