Question

3 any property that is true for \\( \triangle a b c \\) will also be true for \\( \triangle a b c \\) definition of congruence
4 let \\( r, s \\), and \\( t \\) be real numbers such that the vertices of \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\) are defining constants
\\( a ^ { \prime } ( 0,0 ), b ^ { \prime } ( 2 r, 2 s ) \\), and \\( c ^ { \prime } ( 2 t, 0 ) \\)
5 let \\( d ^ { \prime }, e ^ { \prime } \\), and \\( f ^ { \prime } \\) be the midpoints of \\( \overline { a ^ { \prime } b ^ { \prime } }, \overline { b ^ { \prime } c ^ { \prime } } \\) and \\( \overline { a ^ { \prime } c ^ { \prime } } \\) respectively
6 \\( d ^ { \prime } = ( r, s ) \\) defining points
\\( e ^ { \prime } = ( r + t, s ) \\)
\\( f ^ { \prime } = ( t, 0 ) \\)
7 slope of \\( \overline { a ^ { \prime } e ^ { \prime } } = \frac { s } { r + t } \\)
slope of \\( \overline { b ^ { \prime } f ^ { \prime } } = \frac { 2 s } { 2 r - t } \\)
slope of \\( \overline { c ^ { \prime } d ^ { \prime } } = \frac { - s } { 2 t - r } \\) definition of midpoints
8 equation of line \\( \overleftrightarrow { a ^ { \prime } e ^ { \prime } } y = \frac { s } { r + t } \times \\)
equation of line \\( \overleftrightarrow { b ^ { \prime } f ^ { \prime } } y = \frac { 2 s } { 2 r - t } \times + \frac { 2 s t } { 2 r - t } \\)
equation of line \\( \overleftrightarrow { b ^ { \prime } f ^ { \prime } } y = \frac { - s } { 2 t - r } \times - \frac { 2 s t } { 2 t - r } \\) making use of point - slope formula
9 \\( \overleftrightarrow { a ^ { \prime } e ^ { \prime } } \\) and \\( \overleftrightarrow { b ^ { \prime } f ^ { \prime } } \\) intersect at point \\( p \\) algebra
22
what is the reason for statement 7 in the given proof?
\\( \bigcirc \\) a. definition of midpoint
\\( \bigcirc \\) b. definition of slope
\\( \bigcirc \\) c. parallel lines have equal slopes.
\\( \bigcirc \\) d. using point - slope formula

Explanation:

Statement 7 calculates the slopes of line segments. The formula for slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For example, if we have two points \(A'(x_1,y_1)\) and \(E'(x_2,y_2)\), the slope of \(\overline{A'E'}\) is calculated using this formula. This is based on the definition of slope.

Answer:

B. definition of slope