# 7.3: Mistari ya Perp

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##### Ufafanuzi: Perpendicular

Mstari miwili tofauti$$l$$ na$$q$$ ni perpendicular$$l ⊥ q$$, imeandikwa, ikiwa makutano yao huunda pembe nne za kulia au pembe kwa kipimo$$90^{\circ}$$. Miteremko ya mistari ya perpendicular$$l$$ na$$q$$ ni hasi za usawa. Hiyo ni,

$m_l = −\dfrac{1}{m_q} \nonumber$

na

$m_q = − \dfrac{1}{m_l} \nonumber$

##### Mfano Template:index

Kuamua kama mistari iliyotolewa ni perpendicular. Mstari$$l$$ unaopita kupitia pointi$$(0, 1)$$ na$$(1, 3)$$, na mstari$$q$$ unaopita kupitia pointi$$(−1, 4)$$ na$$(5, 1)$$.

Suluhisho

Kuamua kama mistari ni perpendicular, kwanza kupata mteremko wao kwa kutumia mteremko wa formula line. Mteremko wa mstari$$l$$$$m_l$$,, kwamba hupita kupitia pointi$$(0, 1)$$ na$$(1, 3)$$ ni,

$$\begin{array}s m_l &= \dfrac{3 − 1}{1 − 0} \\ &= \dfrac{2}{1} \\ &= 2 \end{array}$$

mteremko wa mstari$$q$$,$$m_q$$, kwamba hupita kwa njia ya pointi$$(−1, 4)$$ na$$(5, 1)$$, ni

$$\begin{array}s m_q &= \dfrac{1 − 4}{5 − (-1)} \\ &= \dfrac{-3}{6} \\ &= \dfrac{-1}{2} \end{array}$$

Sasa, mistari$$l$$ na$$q$$ ni perpendicular kama na tu kama:

$$m_l = −\dfrac{1}{m_q} \text{ and } m_q = −\dfrac{1}{m_l}$$

$$m_l = 2$$na$$m_q = −\dfrac{1}{m_l} = −\dfrac{1}{2}$$. Kwa hiyo, mteremko wa mistari ni usawa mbaya, hivyo inaweza kuhitimishwa kuwa mistari$$l$$ na$$q$$ ni mistari ya perpendicular.

##### Mfano Template:index

Pata mteremko wa mstari perpendicular kwa mstari$$l$$ unaopita kupitia pointi$$(−3, 0)$$ na$$(3, 4)$$.

Suluhisho

Anza kwa kutafuta mteremko wa mstari$$l$$ unaopita kupitia pointi$$(−3, 0)$$ na$$(3, 4)$$, kwa kutumia mteremko wa fomu ya mstari. Hivyo,

$$\begin{array} s m_l &= \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{4 − 0}{3 − (−3)} \\ &= \dfrac{4}{6} \\ &= \dfrac{2}{3} \end{array}$$

Mstari wowote unaozingatia mstari$$l$$ lazima uwe na mteremko ambao ni hasi usawa wa mteremko wake. $$m_l = \dfrac{2}{3}$$Tangu wakati huo mteremko wa mstari perpendicular line$$l$$ lazima$$m = −\dfrac{3}{2}$$

##### Zoezi Template:index

Kuamua kama mistari iliyotolewa ni perpendicular.

1. Mstari$$l$$ unaopita kupitia pointi$$(0, 4)$$$$(5, 3)$$ na na mstari$$q$$ unaopita kupitia pointi$$(1, 5)$$ na$$(−1, −5)$$.
2. Mstari$$l$$ unaopita kupitia pointi$$(−2, −5)$$$$(1, 7)$$ na na mstari$$q$$ unaopita kupitia pointi$$(−4, 1)$$ na$$(−3, −3)$$.
##### Zoezi Template:index

Pata mteremko wa mstari perpendicular kwa:

1. Line$$l$$ kwamba hupita kwa njia ya pointi$$(4, 2)$$ na$$(−1, −2)$$.
2. Line$$q$$ kwamba hupita kwa njia ya pointi$$(7, −8)$$ na$$(9, 1)$$.