Answer:
Proof below
Step-by-step explanation:
Right Triangles
In any right triangle, i.e., where one of its internal angles is 90°, some interesting relations stand. One of the most-used is Pythagora's Theorem.
In a right triangle with shorter sides a and b, and longest side c, called the hypotenuse, the following equation is satisfied:
[tex]c^2=a^2+b^2[/tex]
The image provided in the question shows a line passing through points A(0,4) and B(3,0) that forms a right triangle with both axes.
The origin is marked as C(0,0) and the point M is the midpoint of the segment AB. We have to prove.
[tex]CM=\frac{1}{2}AB[/tex]
First, find the coordinates of the midpoint M(xm,ym):
[tex]\displaystyle x_m=\frac{0+3}{2}=1.5[/tex]
[tex]\displaystyle y_m=\frac{4+0}{2}=2[/tex]
Thus, the midpoint is M( 1.5 , 2 )
Calculate the distance CM:
[tex]CM=\sqrt{(1.5-0)^2+(2-0)^2}[/tex]
[tex]CM=\sqrt{2.25+4}=\sqrt{6.25}=2.5[/tex]
CM=2.5
Now find the distance AB:
[tex]AB=\sqrt{(3-0)^2+(4-0)^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}[/tex]
AB=5
AB/2=2.5
It's proven CM is half of AB
