Question

two points on line m are a(0, 0) and c(b, c). so, the slope of line m is. two points on line n are a(0, 0) and d(b, -d). so, the slope of line n is. the slope of m is equal to the opposite reciprocal of the slope of n, which means. further, since ab = b, bc = c, and bd = d, by substitution. also, cd is perpendicular to the x - axis. since perpendicular lines form right angles and all right angles are congruent, ∠abc and ∠dba are congruent. so, △abc and △dba are similar by the. since corresponding angles in similar triangles are congruent, m∠acb = m∠dab. the acute angles in a right triangle are complementary, so. by substitution,. by the angle addition postulate,. therefore, by substitution. so, line m is perpendicular to line n.

Explanation:

Step1: Find slope of line m

The slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(0,0)$ and $C(b,c)$ on line $m$, the slope of line $m$, $m_m=\frac{c - 0}{b - 0}=\frac{c}{b}$ ($b
eq0$).

Step2: Find slope of line n

For points $A(0,0)$ and $D(b,-d)$ on line $n$, using the slope formula, the slope of line $n$, $m_n=\frac{-d - 0}{b - 0}=-\frac{d}{b}$ ($b
eq0$).

Step3: Check slope - relationship

Since the slope of $m$ is the opposite reciprocal of the slope of $n$, we have $\frac{c}{b}=-\frac{1}{-\frac{d}{b}}=\frac{b}{d}$, so $c\cdot d = b^2$.

Step4: Similar - triangles reason

$\triangle ABC$ and $\triangle DBA$ have $\angle ABC=\angle DBA = 90^{\circ}$ (because $\overline{CD}$ is perpendicular to the x - axis) and $\angle ACB=\angle DAB$ (by similarity of triangles). Triangles are similar by the AA (angle - angle) similarity criterion.

Step5: Complementary - angles property

In right - triangle $\triangle ABC$, $\angle CAB+\angle ACB = 90^{\circ}$, and in right - triangle $\triangle DBA$, $\angle ADB+\angle DAB = 90^{\circ}$.

Step6: Substitution and angle - addition

Since $\angle ACB=\angle DAB$, we can substitute to get $\angle CAB+\angle DAB = 90^{\circ}$. By the angle - addition postulate, $\angle CAD = 90^{\circ}$, which means line $m$ is perpendicular to line $n$.

Answer:

The slope of line $m$ is $\frac{c}{b}$, the slope of line $n$ is $-\frac{d}{b}$, the relationship $\frac{c}{b}=-\frac{1}{-\frac{d}{b}}$ implies $c\cdot d = b^2$, $\triangle ABC$ and $\triangle DBA$ are similar by AA similarity criterion, in right - triangles angles are complementary, substitution gives $\angle CAB+\angle DAB = 90^{\circ}$, and by angle - addition postulate $\angle CAD = 90^{\circ}$ so line $m$ is perpendicular to line $n$.