Find the perimeter of the pentagon MNPQR with vertices ​M(2​, 4​), ​N(5​, 8​), ​P(​8, 4​), ​Q(8​, 1​), and ​R(2​, 1​)

Answer:

The pentagon MNPQR has a perimeter of 22 units.

Step-by-step explanation:

Geometrically speaking, the perimeter of the pentagon is the sum of the lengths of each side, that is:

[tex]p = MN + NP + PQ + QR + RM[/tex] (1)

[tex]p = \sqrt{\overrightarrow{MN}\,\bullet \, \overrightarrow{MN}} + \sqrt{\overrightarrow{NP}\,\bullet \, \overrightarrow{NP}} + \sqrt{\overrightarrow{PQ}\,\bullet \, \overrightarrow{PQ}} + \sqrt{\overrightarrow{QR}\,\bullet \, \overrightarrow{QR}} + \sqrt{\overrightarrow{RM}\,\bullet \, \overrightarrow{RM}}[/tex] (1b)

If we know that [tex]M(x,y) = (2,4)[/tex], [tex]N(x,y) = (5,8)[/tex], [tex]P(x,y) = (8,4)[/tex], [tex]Q(x,y) = (8,1)[/tex] and [tex]R(x,y) = (2,1)[/tex], then the perimeter of the pentagon MNPQR is:

[tex]p =\sqrt{(5-2)^{2}+(8-4)^{2}} + \sqrt{(8-5)^{2}+(4-8)^{2}}+\sqrt{(8-8)^{2}+(1-4)^{2}}+\sqrt{(2-8)^{2}+(1-1)^{2}}+\sqrt{(2-2)^{2}+(4-1)^{2}}[/tex][tex]p = \sqrt{3^{2}+4^{2}} + \sqrt{3^{2}+(-4)^{2}}+\sqrt{0^{2}+(-3)^{2}}+\sqrt{(-6)^{2}+0^{2}}+\sqrt{0^{2}+3^{2}}[/tex]

[tex]p = 22[/tex]

The pentagon MNPQR has a perimeter of 22 units.

The pentagon MNPQR has a perimeter of 22 units.

Geometrically speaking, the perimeter of the pentagon is the sum of the lengths of each side, that is:

p = MN + NP + PQ + QR + RMp=MN+NP+PQ+QR+RM (1)

p = \sqrt{\overrightarrow{MN}\,\bullet \, \overrightarrow{MN}} + \sqrt{\overrightarrow{NP}\,\bullet \, \overrightarrow{NP}} + \sqrt{\overrightarrow{PQ}\,\bullet \, \overrightarrow{PQ}} + \sqrt{\overrightarrow{QR}\,\bullet \, \overrightarrow{QR}} + \sqrt{\overrightarrow{RM}\,\bullet \, \overrightarrow{RM}}p=

MN

∙

MN

+

NP

∙

NP

+

PQ

∙

PQ

+

QR

∙

QR

+

RM

∙

RM

(1b)

If we know that M(x,y) = (2,4)M(x,y)=(2,4) , N(x,y) = (5,8)N(x,y)=(5,8) , P(x,y) = (8,4)P(x,y)=(8,4) , Q(x,y) = (8,1)Q(x,y)=(8,1) and R(x,y) = (2,1)R(x,y)=(2,1) , then the perimeter of the pentagon MNPQR is:

p =\sqrt{(5-2)^{2}+(8-4)^{2}} + \sqrt{(8-5)^{2}+(4-8)^{2}}+\sqrt{(8-8)^{2}+(1-4)^{2}}+\sqrt{(2-8)^{2}+(1-1)^{2}}+\sqrt{(2-2)^{2}+(4-1)^{2}}p=

(5−2)

2

+(8−4)

2

+

(8−5)

2

+(4−8)

2

+

(8−8)

2

+(1−4)

2

+

(2−8)

2

+(1−1)

2

+

(2−2)

2

+(4−1)

2

p = \sqrt{3^{2}+4^{2}} + \sqrt{3^{2}+(-4)^{2}}+\sqrt{0^{2}+(-3)^{2}}+\sqrt{(-6)^{2}+0^{2}}+\sqrt{0^{2}+3^{2}}p=

3

2

+4

2

+

3

2

+(−4)

2

+

0

2

+(−3)

2

+

(−6)

2

+0

2

+

0

2

+3

2

p = 22p=22

The pentagon MNPQR has a perimeter of 22 units.

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