Answer:
DE = m unit , Slope DE = 0 , Mid point of DE = ![(\frac{m}{2},n)](https://tex.z-dn.net/?f=%28%5Cfrac%7Bm%7D%7B2%7D%2Cn%29)
EF = n unit , Slope EF = Undefined , Mid point of EF = ![(m,\frac{n}{2})](https://tex.z-dn.net/?f=%28m%2C%5Cfrac%7Bn%7D%7B2%7D%29)
DF = √(m²+n²) unit , Slope DF = -n / m, Mid point of DF = ![(\frac{m}{2},\frac{n}{2})](https://tex.z-dn.net/?f=%28%5Cfrac%7Bm%7D%7B2%7D%2C%5Cfrac%7Bn%7D%7B2%7D%29)
Step-by-step explanation:
Given:
point D( x₁ , y₁) ≡ ( 0 ,n)
point E( x₂ , y₂) ≡ (m , n)
point F( x₃ , y₃) ≡ ( m ,0)
To Find:
DE = ? , Slope DE = ? , Mid point of DE = ?
EF = ? , Slope EF = ? , Mid point of EF = ?
DF = ? , Slope DF = ? , Mid point of DF = ?
Solution:
We will use Distance Formula,Slope Formula, and Section Formula.
Distance Formula:
![l(DE) = \sqrt{((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} )}](https://tex.z-dn.net/?f=l%28DE%29%20%3D%20%5Csqrt%7B%28%28x_%7B2%7D-x_%7B1%7D%29%5E%7B2%7D%2B%28y_%7B2%7D-y_%7B1%7D%29%5E%7B2%7D%20%29%7D)
![l(DE) = \sqrt{(m-0)^{2}+(n-n)^{2} )}\\l(DE) = \sqrt{(m-0)^{2}+(0)^{2} )}\\l(DE) = \sqrt{(m)^{2} }\\l(DE) = m\ unit](https://tex.z-dn.net/?f=l%28DE%29%20%3D%20%5Csqrt%7B%28m-0%29%5E%7B2%7D%2B%28n-n%29%5E%7B2%7D%20%29%7D%5C%5Cl%28DE%29%20%3D%20%5Csqrt%7B%28m-0%29%5E%7B2%7D%2B%280%29%5E%7B2%7D%20%29%7D%5C%5Cl%28DE%29%20%3D%20%5Csqrt%7B%28m%29%5E%7B2%7D%20%7D%5C%5Cl%28DE%29%20%3D%20m%5C%20unit)
Similarly,
![l(EF) = \sqrt{((x_{3}-x_{2})^{2}+(y_{3}-y_{2})^{2} )}\\l(EF) = \sqrt{((m-m)^{2}+(0-n)^{2} )}\\l(EF) = \sqrt{((0)^{2}+(-n)^{2} )}\\l(EF) = \sqrt{(n)^{2}}\\l(EF) = n\ unit](https://tex.z-dn.net/?f=l%28EF%29%20%3D%20%5Csqrt%7B%28%28x_%7B3%7D-x_%7B2%7D%29%5E%7B2%7D%2B%28y_%7B3%7D-y_%7B2%7D%29%5E%7B2%7D%20%29%7D%5C%5Cl%28EF%29%20%3D%20%5Csqrt%7B%28%28m-m%29%5E%7B2%7D%2B%280-n%29%5E%7B2%7D%20%29%7D%5C%5Cl%28EF%29%20%3D%20%5Csqrt%7B%28%280%29%5E%7B2%7D%2B%28-n%29%5E%7B2%7D%20%29%7D%5C%5Cl%28EF%29%20%3D%20%5Csqrt%7B%28n%29%5E%7B2%7D%7D%5C%5Cl%28EF%29%20%3D%20n%5C%20unit)
Similarly,
![l(DF) = \sqrt{((x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2} )}\\l(DF) = \sqrt{((m-0)^{2}+(0-n)^{2} )}\\l(DF) = \sqrt{((m)^{2}+(n)^{2} )}\ units](https://tex.z-dn.net/?f=l%28DF%29%20%3D%20%5Csqrt%7B%28%28x_%7B3%7D-x_%7B1%7D%29%5E%7B2%7D%2B%28y_%7B3%7D-y_%7B1%7D%29%5E%7B2%7D%20%29%7D%5C%5Cl%28DF%29%20%3D%20%5Csqrt%7B%28%28m-0%29%5E%7B2%7D%2B%280-n%29%5E%7B2%7D%20%29%7D%5C%5Cl%28DF%29%20%3D%20%5Csqrt%7B%28%28m%29%5E%7B2%7D%2B%28n%29%5E%7B2%7D%20%29%7D%5C%20units)
Slope Formula:
![Slope(DE)=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\\\\Slope(DE)=\frac{n-n}{m-0}\\\\Slope(DE)=0](https://tex.z-dn.net/?f=Slope%28DE%29%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%20%7D%7Bx_%7B2%7D-x_%7B1%7D%20%7D%5C%5C%5C%5CSlope%28DE%29%3D%5Cfrac%7Bn-n%7D%7Bm-0%7D%5C%5C%5C%5CSlope%28DE%29%3D0)
Similarly,
![Slope(EF)=\frac{y_{3}-y_{2} }{x_{3}-x_{2}}\\\\Slope(EF)=\frac{0-n}{m-m}\\\\Slope(EF)=Infinity](https://tex.z-dn.net/?f=Slope%28EF%29%3D%5Cfrac%7By_%7B3%7D-y_%7B2%7D%20%7D%7Bx_%7B3%7D-x_%7B2%7D%7D%5C%5C%5C%5CSlope%28EF%29%3D%5Cfrac%7B0-n%7D%7Bm-m%7D%5C%5C%5C%5CSlope%28EF%29%3DInfinity)
Similarly,
![Slope(DF)=\frac{y_{3}-y_{1} }{x_{3}-x_{1} }\\\\Slope(DE)=\frac{0-n}{m-0}\\\\Slope(DE)=\frac {-n}{m}](https://tex.z-dn.net/?f=Slope%28DF%29%3D%5Cfrac%7By_%7B3%7D-y_%7B1%7D%20%7D%7Bx_%7B3%7D-x_%7B1%7D%20%7D%5C%5C%5C%5CSlope%28DE%29%3D%5Cfrac%7B0-n%7D%7Bm-0%7D%5C%5C%5C%5CSlope%28DE%29%3D%5Cfrac%20%7B-n%7D%7Bm%7D)
Section Formula:
![Mid\ point(DE)=(\frac{x_{1}+x_{2} }{2}, \frac{y_{1}+y_{2} }{2})=(\frac{m}{2},n)](https://tex.z-dn.net/?f=Mid%5C%20point%28DE%29%3D%28%5Cfrac%7Bx_%7B1%7D%2Bx_%7B2%7D%20%7D%7B2%7D%2C%20%5Cfrac%7By_%7B1%7D%2By_%7B2%7D%20%7D%7B2%7D%29%3D%28%5Cfrac%7Bm%7D%7B2%7D%2Cn%29)
Similarly,
![Mid\ point(EF)=(\frac{x_{3}+x_{2} }{2}, \frac{y_{3}+y_{2} }{2})=(m,\frac{n}{2})](https://tex.z-dn.net/?f=Mid%5C%20point%28EF%29%3D%28%5Cfrac%7Bx_%7B3%7D%2Bx_%7B2%7D%20%7D%7B2%7D%2C%20%5Cfrac%7By_%7B3%7D%2By_%7B2%7D%20%7D%7B2%7D%29%3D%28m%2C%5Cfrac%7Bn%7D%7B2%7D%29)
Similarly,
![Mid\ point(DE)=(\frac{x_{1}+x_{3} }{2}, \frac{y_{1}+y_{3} }{2})=(\frac{m}{2},\frac{n}{2})](https://tex.z-dn.net/?f=Mid%5C%20point%28DE%29%3D%28%5Cfrac%7Bx_%7B1%7D%2Bx_%7B3%7D%20%7D%7B2%7D%2C%20%5Cfrac%7By_%7B1%7D%2By_%7B3%7D%20%7D%7B2%7D%29%3D%28%5Cfrac%7Bm%7D%7B2%7D%2C%5Cfrac%7Bn%7D%7B2%7D%29)
DE = m unit , Slope DE = 0 , Mid point of DE = ![(\frac{m}{2},n)](https://tex.z-dn.net/?f=%28%5Cfrac%7Bm%7D%7B2%7D%2Cn%29)
EF = n unit , Slope EF = Undefined , Mid point of EF = ![(m,\frac{n}{2})](https://tex.z-dn.net/?f=%28m%2C%5Cfrac%7Bn%7D%7B2%7D%29)
DF = √(m²+n²) unit , Slope DF = -n / m, Mid point of DF = ![(\frac{m}{2},\frac{n}{2})](https://tex.z-dn.net/?f=%28%5Cfrac%7Bm%7D%7B2%7D%2C%5Cfrac%7Bn%7D%7B2%7D%29)