Mathematics

Question

Solve the problems below. Please answer with completely simplified exact value(s) or expression(s). Given: ΔАВС, m∠ACB = 90 CD ⊥ AB , m∠ACD = 30°,AD = 8 cm. Find: Perimeter of ΔABC
Solve the problems below. Please answer with completely simplified exact value(s) or expression(s). Given: ΔАВС, m∠ACB = 90 CD   ⊥  AB   , m∠ACD = 30°,AD = 8 cm

1 Answer

  • Answer:

    Perimeter is 75.7128129211 units

    Step-by-step explanation:

    Given ΔАВС, m∠ACB = 90°, CD ⊥ AB and m∠ACD = 30, AD = 8 cm

    we have to find the perimeter of ΔABC

    In triangle ADC,

    [tex]\sin 30^{\circ}=\frac{AD}{AC}=\frac{8}{AC}[/tex]

    [tex]\frac{1}{2}=\frac{8}{AC}[/tex] ⇒ [tex]AC=16units[/tex]

    and also, [tex]\tan 30^{\circ}=\frac{AD}{CD}=\frac{8}{CD}[/tex]

    [tex]\frac{1}{\sqrt{3}}=\frac{8}{CD}[/tex] ⇒ [tex]CD=8\sqrt{3}units[/tex]

    Now, in triangle BDC,

    ∠BDC + ∠ADC = 180°

    ∠BDC = 180°- 90° = 90°

    and also ∠DCB=∠ACB - ∠ACD = 90° - 30° = 60°

    [tex]\tan 60^{\circ}=\frac{DB}{CD}=\frac{DB}{8\sqrt{3} }[/tex]

    DB=[tex]{\sqrt{3}}\times{8}{\sqrt{3}}[/tex] ⇒ [tex]DB=24units[/tex]

    and also [tex]\sin 60^{\circ}=\frac{DB}{BC}=\frac{24}{BC}[/tex]

    [tex]\frac{\sqrt{3}}{2}=\frac{24}{BC}[/tex] ⇒ [tex]BC=\frac{48}{\sqrt{3}} units[/tex]

    Hence, Perimeter = AC+AD+DB+BC

                                 = 16+8+24+[tex]\frac{48}{\sqrt{3} }[/tex]

                                = 75.7128129211 units



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