Euclid3

Calculate the height and sides of the right triangle if one leg is a = 100 km and the section of hypotenuse adjacent to the second leg cb = 14 km.

Correct answer:

b =  68.5586 km
c =  121.2447 km
h =  38.7482 km

Step-by-step explanation:

a=100 km c2=14 km a2 = c c1 b2 = c c2  c = a2:c1 = b2 : c2 c=c1+c2  a2 = c (cc2)  a2=x (xc2)  1002=x (x14) x2+14x+10000=0 x214x10000=0  a=1;b=14;c=10000 D=b24ac=14241(10000)=40196 D>0  x1,2=b±D2a=14±401962=14±2100492 x1,2=7±100.244701 x1=107.244700608 x2=93.244700608  c1=x1=107.2447107.2447 km  c=c1+c2=107.2447+14121.2447 km  b=c2a2=121.244721002=68.5586 kma = 100 \ \text{km} \ \\ c_{2} = 14 \ \text{km} \ \\ a^2\ = \ c \cdot \ c_{1} \ \\ b^2\ = \ c \cdot \ c_{2} \ \\ \ \\ c\ = \ a^2:c_{1}\ = \ b^2\ :\ c_{2} \ \\ c = c_{1}+c_{2} \ \\ \ \\ a^2\ = \ c \cdot \ (c-c_{2}) \ \\ \ \\ a^2 = x \cdot \ (x-c_{2}) \ \\ \ \\ 100^2 = x \cdot \ (x-14) \ \\ -x^2 +14x +10000 = 0 \ \\ x^2 -14x -10000 = 0 \ \\ \ \\ a = 1; b = -14; c = -10000 \ \\ D = b^2 - 4ac = 14^2 - 4 \cdot 1 \cdot (-10000) = 40196 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 14 \pm \sqrt{ 40196 } }{ 2 } = \dfrac{ 14 \pm 2 \sqrt{ 10049 } }{ 2 } \ \\ x_{1,2} = 7 \pm 100.244701 \ \\ x_{1} = 107.244700608 \ \\ x_{2} = -93.244700608 \ \\ \ \\ c_{1} = x_{1} = 107.2447 \doteq 107.2447 \ \text{km} \ \\ \ \\ c = c_{1}+c_{2} = 107.2447+14 \doteq 121.2447 \ \text{km} \ \\ \ \\ b = \sqrt{ c^2-a^2 } = \sqrt{ 121.2447^2-100^2 } = 68.5586 \ \text{km}

Our quadratic equation calculator calculates it.

h=c1 c2=107.2447 1438.7482 km   Verifying Solution:  S1=a b2=100 68.558623427.9285 S2=c h2=121.2447 38.748222349.0091  a2=c c1=121.2447 107.2447114.03 b2=c c2=121.2447 1441.1998 c3=a2+b2=1002+68.55862121.2447h = \sqrt{ c_{1} \cdot \ c_{2} } = \sqrt{ 107.2447 \cdot \ 14 } \doteq 38.7482 \ \text{km} \ \\ \ \\ \text{ Verifying Solution: } \ \\ S_{1} = \dfrac{ a \cdot \ b }{ 2 } = \dfrac{ 100 \cdot \ 68.5586 }{ 2 } \doteq 3427.9285 \ \\ S_{2} = \dfrac{ c \cdot \ h }{ 2 } = \dfrac{ 121.2447 \cdot \ 38.7482 }{ 2 } \doteq 2349.0091 \ \\ \ \\ a_{2} = \sqrt{ c \cdot \ c_{1} } = \sqrt{ 121.2447 \cdot \ 107.2447 } \doteq 114.03 \ \\ b_{2} = \sqrt{ c \cdot \ c_{2} } = \sqrt{ 121.2447 \cdot \ 14 } \doteq 41.1998 \ \\ c_{3} = \sqrt{ a^2+b^2 } = \sqrt{ 100^2+68.5586^2 } \doteq 121.2447



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