📐 Penyelesaian SegitigaSolution of Triangles

Petua Sin (Silang-silang)Sine Rule (Cross)

\(\displaystyle\frac{\bl{a}}{\sin\bl{A}}=\frac{\og{b}}{\sin\og{B}}\)
Guna bila ada sepasang sisi dan sudut bertentangan yang diketahui.Use when a side and its opposite angle are known.

Petua Kosinus (Bunga)Cosine Rule (Flower)

\(\og{c}^2=\bl{a}^2+\bl{b}^2-2\bl{a}\bl{b}\cos\og{\theta}\)
θ ialah sudut yang dikepit oleh sisi a dan b (bertentangan dengan c). Guna bila diberi 2 sisi + sudut kepit, atau 3 sisi.θ is the angle between sides a and b (opposite c). Use when given 2 sides + the included angle, or all 3 sides.

1 sudut dikepit 2 sisi1 angle between 2 sides

\(\text{Luas}=\dfrac{1}{2}\bl{a}\bl{b}\sin\og{\theta}\)Area
Sudut θ mesti di antara dua sisi yang diberi.θ must lie between the two given sides.

Heron

\(s=\dfrac{a+b+c}{2}\)
\(\text{Luas}=\sqrt{s(s-a)(s-b)(s-c)}\)Area
Semua sisi diberiAll sides are given

Petua Sin (Silang-silang)Sine Rule (Cross)

\(\displaystyle\frac{\bl{a}}{\sin\bl{A}}=\frac{\og{b}}{\sin\og{B}}\)
Guna bila ada sepasang sisi dan sudut bertentangan yang diketahui.Use when a side and its opposite angle are known.

Cari sisiFind a side

\(b=\dfrac{a\sin B}{\sin A}\)

Cari sudutFind an angle

\(\sin B=\dfrac{b\sin A}{a}\)

Simulasi: ubah sudut dan sisiSimulation: change the angles and side

∠A
∠B
a

Petua Kosinus (Bunga)Cosine Rule (Flower)

\(\og{c}^2=\bl{a}^2+\bl{b}^2-2\bl{a}\bl{b}\cos\og{\theta}\)
θ ialah sudut yang dikepit oleh sisi a dan b (bertentangan dengan c). Guna bila diberi 2 sisi + sudut kepit, atau 3 sisi.θ is the angle between sides a and b (opposite c). Use when given 2 sides + the included angle, or all 3 sides.

Cari sudutFind an angle

\(\cos\theta=\dfrac{a^2+b^2-c^2}{2ab}\)

Simulasi: ubah sisi dan sudutSimulation: change the sides and angle

a
b
θ

LuasArea

1 sudut dikepit 2 sisi1 angle between 2 sides
\(\text{Luas}=\dfrac{1}{2}\bl{a}\bl{b}\sin\og{\theta}\)Area
Sudut θ mesti di antara dua sisi yang diberi.θ must lie between the two given sides.

Simulasi: ubah sisi dan sudutSimulation: change the sides and angle

a
b
θ

HeronHeron's Formula

Semua sisi diberiAll sides are given
\(s=\dfrac{a+b+c}{2}\)
\(\text{Luas}=\sqrt{s(s-a)(s-b)(s-c)}\)Area

Cara gunaHow to use

  • Cari s (setengah perimeter) dahulu.Find s (the semi-perimeter) first.
  • Gantikan ke dalam formula, kemudian punca kuasa dua.Substitute into the formula, then take the square root.

Contoh (6, 7, 9 cm)Example (6, 7, 9 cm)

\(s=\dfrac{7+6+9}{2}=11\)
\(\text{Luas}=\sqrt{11(11-7)(11-6)(11-9)}\)
\(=\sqrt{11(4)(5)(2)}=\sqrt{440}\approx 20.98\ \text{cm}^2\)Area ≈ 20.98 cm²

Simulasi: ubah sisiSimulation: change the sides

a
b
c

Kes AmbiguitiAmbiguous Cases

2 fix, 1 change

Diberi 3 info (SSA)Given 3 info (SSA)

  • Lakarkan sudutSketch the angle
  • Lakarkan sisi yang berkait dengan sudutSketch the side corresponding to the angle
  • Lakarkan sisi yang satu lagiSketch the other side
\(CB=C'B'\qquad AC=A'C'\qquad \angle CAB=\angle C'A'B'\)
Tetapkan sudut A dan sisi AC (2 fix), ubah panjang CB (1 change). Busur berpusat C boleh memotong garis AB pada dua titik.Fix angle A and side AC (2 fix), change the length of CB (1 change). The arc centred at C can cut line AB at two points.

Diberi 2 infoGiven 2 info

  • Panjangkan sisiExtend the sides
  • Ubah kedudukan sudutChange the position of the angle
Titik B′ terletak pada garis AB.Point B′ lies on line AB.
\(CB=C'B'\;\Rightarrow\;\sin\angle ABC=\sin\angle A'B'C'\)
\(\angle AB'C=180^\circ-\angle ABC\)

Simulasi: ubah CBSimulation: change CB

∠A
AC
CB