# Triangle

Triangle KLM is given by plane coordinates of vertices: K[11, -10] L[10, 12] M[1, 3].

Calculate its area and itsinterior angles.

Result

S =  103.5
K =  34.966 °
L =  47.603 °
M =  97.431 °

#### Solution:

$x_{ 0 } = 11 \ \\ y_{ 0 } = -10 \ \\ \ \\ x_{ 1 } = 10 \ \\ y_{ 1 } = 12 \ \\ \ \\ x_{ 2 } = 1 \ \\ y_{ 2 } = 3 \ \\ \ \\ \ \\ LM = M-L = (k_{ 0 },k_{ 1 }) \ \\ k_{ 0 } = x_{ 2 }-x_{ 1 } = 1-10 = -9 \ \\ k_{ 1 } = y_{ 2 }-y_{ 1 } = 3-12 = -9 \ \\ \ \\ KM = M-K = (l_{ 0 },l_{ 1 }) \ \\ l_{ 0 } = x_{ 2 }-x_{ 0 } = 1-11 = -10 \ \\ l_{ 1 } = y_{ 2 }-y_{ 0 } = 3-(-10) = 13 \ \\ \ \\ LK = K-L = (m_{ 0 },m_{ 1 }) \ \\ m_{ 0 } = x_{ 0 }-x_{ 1 } = 11-10 = 1 \ \\ m_{ 1 } = y_{ 0 }-y_{ 1 } = (-10)-12 = -22 \ \\ \ \\ \ \\ k = \sqrt{ k_{ 0 }^2+k_{ 1 }^2 } = \sqrt{ (-9)^2+(-9)^2 } = 9 \ \sqrt{ 2 } \doteq 12.7279 \ \\ l = \sqrt{ l_{ 0 }^2+l_{ 1 }^2 } = \sqrt{ (-10)^2+13^2 } = \sqrt{ 269 } \doteq 16.4012 \ \\ m = \sqrt{ m_{ 0 }^2+m_{ 1 }^2 } = \sqrt{ 1^2+(-22)^2 } = \sqrt{ 485 } \doteq 22.0227 \ \\ \ \\ s = \dfrac{ k+l+m }{ 2 } = \dfrac{ 12.7279+16.4012+22.0227 }{ 2 } \doteq 25.5759 \ \\ S = \sqrt{ s \cdot \ (s-k) \cdot \ (s-l) \cdot \ (s-m) } = \sqrt{ 25.5759 \cdot \ (25.5759-12.7279) \cdot \ (25.5759-16.4012) \cdot \ (25.5759-22.0227) } = \dfrac{ 207 }{ 2 } = 103.5$
$K = angle(KL, KM) \ \\ K_{ 1 } = \arccos(\dfrac{ -l_{ 0 } \cdot \ m_{ 0 }-l_{ 1 } \cdot \ m_{ 1 } }{ l \cdot \ m } ) = \arccos(\dfrac{ -(-10) \cdot \ 1-13 \cdot \ (-22) }{ 16.4012 \cdot \ 22.0227 } ) \doteq 0.6103 \ \\ K = K_{ 1 } \rightarrow \ ^\circ = K_{ 1 } \cdot \ \dfrac{ 180 }{ \pi } \ \ ^\circ = 34.9660298264 \ \ ^\circ = 34.966 ^\circ = 34^\circ 57'58"$
$L = angle(LK, LM) \ \\ L_{ 1 } = \arccos(\dfrac{ k_{ 0 } \cdot \ m_{ 0 }+k_{ 1 } \cdot \ m_{ 1 } }{ k \cdot \ m } ) = \arccos(\dfrac{ (-9) \cdot \ 1+(-9) \cdot \ (-22) }{ 12.7279 \cdot \ 22.0227 } ) \doteq 0.8308 \ \\ L = L_{ 1 } \rightarrow \ ^\circ = L_{ 1 } \cdot \ \dfrac{ 180 }{ \pi } \ \ ^\circ = 47.6025622026 \ \ ^\circ = 47.603 ^\circ = 47^\circ 36'9"$
$M_{ 1 } = \arccos(\dfrac{ k_{ 0 } \cdot \ l_{ 0 }+k_{ 1 } \cdot \ l_{ 1 } }{ k \cdot \ l } ) = \arccos(\dfrac{ (-9) \cdot \ (-10)+(-9) \cdot \ 13 }{ 12.7279 \cdot \ 16.4012 } ) \doteq 1.7005 \ \\ M = M_{ 1 } \rightarrow \ ^\circ = M_{ 1 } \cdot \ \dfrac{ 180 }{ \pi } \ \ ^\circ = 97.4314079711 \ \ ^\circ = 97.431 ^\circ = 97^\circ 25'53"$

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#### Following knowledge from mathematics are needed to solve this word math problem:

For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc. Two vectors given by its magnitudes and by included angle can be added by our vector sum calculator. Cosine rule uses trigonometric SAS triangle calculator. See also our trigonometric triangle calculator. Pythagorean theorem is the base for the right triangle calculator.

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