Question

proof:
find the length of \\( \overline { c m } \\)
\\( \

$$\begin{aligned} c m & = \\sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } } \\\\ & = \\sqrt { ( 1.5 - 0 ) ^ { 2 } + ( 2 - 0 ) ^ { 2 } } \\\\ & = \\sqrt { ( 1.5 ) ^ { 2 } + 2 ^ { 2 } } \\\\ & = \\sqrt { 2.25 + 4 } = \\sqrt { 6.25 } = \\end{aligned}$$

\\)
find the length of \\( \overline { a b } \\)
\\( \

$$\begin{aligned} a b & = \\sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } } \\\\ & = \\sqrt { ( 3 - 0 ) ^ { 2 } + ( 0 - 4 ) ^ { 2 } } \\\\ & = \\sqrt { 3 ^ { 2 } + ( - 4 ) ^ { 2 } } \\\\ & = \\sqrt { 9 + 16 } = \\sqrt { 25 } = \\end{aligned}$$

\\)
therefore:

\\( \

$$\begin{array} { l } { \\text { a. } a b = \\frac { 1 } { 2 } c m } \\\\ { \\text { b. } c m = \\frac { 1 } { 2 } a b } \\\\ { \\text { c. } 5 } \\\\ { \\text { d. } 2.5 } \\\\ { \\text { e. } 3.125 } \\end{array}$$

\\)

Explanation:

Step1: Calculate the length of \( \overline{CM} \)

Given \( C(0,0) \) and \( M(1.5,2) \), use the distance formula \( CM=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
Substitute \( x_1 = 0,y_1 = 0,x_2=1.5,y_2 = 2 \) into the formula:
\( CM=\sqrt{(1.5 - 0)^2+(2 - 0)^2}=\sqrt{2.25 + 4}=\sqrt{6.25}=2.5 \)

Step2: Calculate the length of \( \overline{AB} \)

Given \( A(0,4) \) and \( B(3,0) \), use the distance formula \( AB=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
Substitute \( x_1 = 0,y_1 = 4,x_2 = 3,y_2 = 0 \) into the formula:
\( AB=\sqrt{(3 - 0)^2+(0 - 4)^2}=\sqrt{9 + 16}=\sqrt{25}=5 \)

Answer:

\( CM = 2.5 \), \( AB=5 \), \( CM=\frac{1}{2}AB \) (Option b)