ຮູບສາມແຈ ແມ່ນ ຮູບຮ່າງ ພື້ນຖານໜຶ້່ງ ໃນ ເລຂາຄະນິດ : ແມ່ນ ຮູບຫຼາຍແຈ ທີ່ມີ 3 ແຈ ແລະ ສາມຂ້າງ ທີ່ຕິດຕໍ່ກັນ ດ້ວຍເສັ້ນ (ຄະນິດສາດ) . ຮູບສາມແຈ ທີ່ ປະກອບດ້ວຍ ມຸມ A , B , ແລະ C ຈະສາມາດສະແດງໄດ້ ດ້ວຍເຄື່ອງໝາຍ △ABC.ໃນ ເລຂາຄະນິດຢູຄລິດ ສາມ ເມັດ ທີ່ບໍ່ນອນໃນເສັ້ນຊື່ດຽວກັນໃດໜຶ່ງ ຈະປະກອບເປັນ ຮູບສາມແຈ ທີ່ແນ່ນອນພຽງອັນດຽວເທົ່ານັ້ນ.
ຮູບສາມແຈ. ຮູບສາມແຈສາມາດ ແບ່ງ ຕາມຄວາມສຳພັນ ຂອງ ຄວາມຍາວແຕ່ລະຂ້າງ:
Equilateral Isosceles Scalene
ຮູບສາມແຈຍັງສາມາດແບ່ງຕາມຂະໜາດຂອງມຸມ:
ສູດຄິດໄລ່ເນື້ອທີ່ຮູບສາມແຈ ແມ່ນ :
S
=
1
2
b
h
{\displaystyle S={\frac {1}{2}}bh}
ໂດຍທີ່
S
{\displaystyle S}
b
{\displaystyle b}
h
{\displaystyle h}
ເນື້ອທີ່ຂອງຮູບສາມແຈ ABC ສາມາດຄິດໄລ່ໄດ້ໂດຍ:
1
2
(
A
B
⋅
A
B
)
(
A
C
⋅
A
C
)
−
(
A
B
⋅
A
C
)
2
=
1
2
|
A
B
|
2
|
A
C
|
2
−
(
A
B
⋅
A
C
)
2
.
{\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}\,.}
ໃຊ້ໄຕມຸມມິຕິ ເພື່ອຄິດໄລ່ ລວງສູງ h . ເນື້ອທີ່ຂອງ ຮູບສາມແຈ ແມ່ນ:
S
=
1
2
a
b
sin
γ
=
1
2
b
c
sin
α
=
1
2
c
a
sin
β
.
{\displaystyle S={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta .}
ແຕ່ເນື່ອງຈາກວ່າ sin α = sin (π - α) = sin (β + γ), ແລະເຊັ່ນດຽວກັນກັບສອງມຸມອື່ນ:
S
=
1
2
a
b
sin
(
α
+
β
)
=
1
2
b
c
sin
(
β
+
γ
)
=
1
2
c
a
sin
(
γ
+
α
)
.
{\displaystyle S={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).}
S
=
1
2
|
det
(
x
B
x
C
y
B
y
C
)
|
=
1
2
|
x
B
y
C
−
x
C
y
B
|
.
{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}
ສຳລັບໂຕປະສານທົ່ວໄປ:
S
=
1
2
|
det
(
x
A
x
B
x
C
y
A
y
B
y
C
1
1
1
)
|
=
1
2
|
x
A
y
C
−
x
A
y
B
+
x
B
y
A
−
x
B
y
C
+
x
C
y
B
−
x
C
y
A
|
{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{C}-x_{A}y_{B}+x_{B}y_{A}-x_{B}y_{C}+x_{C}y_{B}-x_{C}y_{A}{\big |}}
S
=
1
2
|
(
x
C
−
x
A
)
(
y
B
−
y
A
)
−
(
x
B
−
x
A
)
(
y
C
−
y
A
)
|
.
{\displaystyle S={\frac {1}{2}}{\big |}(x_{C}-x_{A})(y_{B}-y_{A})-(x_{B}-x_{A})(y_{C}-y_{A}){\big |}.}
ໃນສາມມິຕິ ເນື້ອທີ່ຂອງຮູບສາມແຈທົ່ວໄປ {A = (x A , y A , z A ), B = (x B , y B , z B ) ແລະ C = (x C , y C , z C )} ແມ່ນ ຜົນບວກປີຕາກໍ ຂອງ ເນື້ອທີ່ເງົາສາຍ ຢູ່ ແຕ່ລະໜ້າພຽງ (i.e. x = 0, y = 0 and z = 0):
S
=
1
2
(
det
(
x
A
x
B
x
C
y
A
y
B
y
C
1
1
1
)
)
2
+
(
det
(
y
A
y
B
y
C
z
A
z
B
z
C
1
1
1
)
)
2
+
(
det
(
z
A
z
B
z
C
x
A
x
B
x
C
1
1
1
)
)
2
.
{\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}
S
=
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}
ໂດຍທີ່ s = ½ (a + b + c ) ແມ່ນ ລວງຮອບຂອງຮູບສາມແຈຫານສອງ..
ວິທີຂຽນສູດເຮຣອນສາມແບບ
S
=
1
4
(
a
2
+
b
2
+
c
2
)
2
−
2
(
a
4
+
b
4
+
c
4
)
{\displaystyle S={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}}
S
=
1
4
2
(
a
2
b
2
+
a
2
c
2
+
b
2
c
2
)
−
(
a
4
+
b
4
+
c
4
)
{\displaystyle S={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
S
=
1
4
(
a
+
b
−
c
)
(
a
−
b
+
c
)
(
−
a
+
b
+
c
)
(
a
+
b
+
c
)
.
{\displaystyle S={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.}
ຮູບສາມແຈ ທີ່ມີຂ້າງ a, b ແລະ c ແລະ ມຸມ α, β ແລະ γ ຕາມລຳດັບ. ຊິນ
a
sin
α
=
b
sin
β
=
c
sin
γ
{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}
ສຳຫຼັບ ຮູບສາມແຈ ທີ່ ລວງຍາວຂ້າງ
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
a
{\displaystyle a}
b
{\displaystyle b}
γ
{\displaystyle \gamma }
c
{\displaystyle c}
c
{\displaystyle c}
c
2
=
a
2
+
b
2
−
2
a
b
cos
(
γ
)
⟹
b
2
=
a
2
+
c
2
−
2
a
c
cos
(
β
)
⟹
a
2
=
b
2
+
c
2
−
2
b
c
cos
(
α
)
{\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )\implies b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )\implies a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )}
ຊິນ
sin
A
=
opposite
hypotenuse
=
a
h
.
{\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}\,.}
ໂກຊິນ
cos
A
=
adjacent
hypotenuse
=
b
h
.
{\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}\,.}
ຕັງຊັງ
tan
A
=
opposite
adjacent
=
a
b
.
{\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}\,.}
ອາກຊິນ
θ
=
arcsin
(
opposite
hypotenuse
)
.
{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right).}
ອາກໂກຊ
θ
=
arccos
(
adjacent
hypotenuse
)
.
{\displaystyle \theta =\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}
ອາກຕັງ
θ
=
arctan
(
opposite
adjacent
)
.
{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right).}
ຈຳນວນຂ້າງ
1-10 ຂ້າງ 11-20 ຂ້າງ
