A quadrilateral has its vertices at points A(-4,3), B(-2,3), C(-2,1), and D(-4,1). a) Plot the points on a cartesian plane and from them, what is the shape of the figure formed?

Nukta za pembe nne ni:

$$A(-4, 3)$$ $$B(-2, 3)$$ $$C(-2, 1)$$ $$D(-4, 1)$$

Ili kuchora nukta hizi, chora mhimili wa x (mlalo) na mhimili wa y (wima) unaokatana kwenye asili (0,0). Kisha, kwa kila nukta $(x, y)$:

• Anza kwenye asili.
• Sogea $x$ vitengo kushoto (kama $x$ ni hasi) au kulia (kama $x$ ni chanya).
• Kutoka hapo, sogea $y$ vitengo chini (kama $y$ ni hasi) au juu (kama $y$ ni chanya).
• Weka alama kwenye eneo hilo. Baada ya kuchora nukta zote nne, ziunganishe kwa mpangilio (A hadi B, B hadi C, C hadi D, na D hadi A) ili kuunda pembe nne.

Tutatumia fomula ya umbali $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Urefu wa AB:

$$AB = \sqrt{(-2 - (-4))^2 + (3 - 3)^2}$$ $$AB = \sqrt{(2)^2 + (0)^2}$$ $$AB = \sqrt{4 + 0}$$ $$AB = \sqrt{4} = 2 \text{ vitengo}$$

Urefu wa BC:

$$BC = \sqrt{(-2 - (-2))^2 + (1 - 3)^2}$$ $$BC = \sqrt{(0)^2 + (-2)^2}$$ $$BC = \sqrt{0 + 4}$$ $$BC = \sqrt{4} = 2 \text{ vitengo}$$

Urefu wa CD:

$$CD = \sqrt{(-4 - (-2))^2 + (1 - 1)^2}$$ $$CD = \sqrt{(-2)^2 + (0)^2}$$ $$CD = \sqrt{4 + 0}$$ $$CD = \sqrt{4} = 2 \text{ vitengo}$$

Urefu wa DA:

$$DA = \sqrt{(-4 - (-4))^2 + (3 - 1)^2}$$ $$DA = \sqrt{(0)^2 + (2)^2}$$ $$DA = \sqrt{0 + 4}$$ $$DA = \sqrt{4} = 2 \text{ vitengo}$$

Tutatumia fomula ya mteremko $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Mteremko wa AB:

$$m_{AB} = \frac{3 - 3}{-2 - (-4)} = \frac{0}{2} = 0$$

(Mstari mlalo)

Mteremko wa BC:

$$m_{BC} = \frac{1 - 3}{-2 - (-2)} = \frac{-2}{0}$$

(Mteremko haujafafanuliwa, mstari wima)

Mteremko wa CD:

$$m_{CD} = \frac{1 - 1}{-4 - (-2)} = \frac{0}{-2} = 0$$

(Mstari mlalo)

Mteremko wa DA:

$$m_{DA} = \frac{3 - 1}{-4 - (-4)} = \frac{2}{0}$$

(Mteremko haujafafanuliwa, mstari wima)

• Tumegundua kuwa pande zote nne (AB, BC, CD, DA) zina urefu sawa wa vitengo 2.
• Mstari AB na CD ni milalo (mteremko 0), hivyo ni sambamba.
• Mstari BC na DA ni wima (mteremko haujafafanuliwa), hivyo ni sambamba.
• Pande zinazopakana (kama AB na BC) ni perpendicular kwa sababu moja ni mlalo na nyingine ni wima.

Kwa kuwa pande zote zina urefu sawa na pembe zote ni pembe mraba (digrii 90), umbo lililoundwa ni mraba.

Umbo la umbo lililoundwa ni: $\text{Mraba}$