# What is the perimeter of a triangle with corners at (7 ,5 ), (8 ,2 ), and (4 ,7 )?

Mar 23, 2018

Perimeter of the triangle = color(blue)(13.171 Units

#### Explanation:

Given:

The three vertices of a triangle is given to us.

Let the vertices be A, B, and C.

Hence, we have color(blue)(A(7, 5), B(8, 2) and C(4, 7) as shown below:

Observe that we have three segments: $\overline{A B} , \overline{B C} \mathmr{and} \overline{A C}$

Find the magnitude of $\overline{A B}$ using the Distance formula:

Distance formula: $A B = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

color(red)("Step 1"

Find the distance between the points:-

color(blue)(A(7, 5) and B(8, 2)

Let us say the points are $A \left({x}_{1} , {y}_{1}\right) \mathmr{and} B \left({x}_{2} , {y}_{2}\right)$

Then, x_1 = 7; x_2=8, y_1 = 5 and y_2=2

Using the distance formula, we get

$\sqrt{{\left(8 - 7\right)}^{2} + {\left(2 - 5\right)}^{2}}$

$\Rightarrow \sqrt{1 + 9}$

$\Rightarrow \sqrt{10}$

$\approx 3.16228$

Hence, color(green)(bar(AB)~~3.16228

color(red)("Step 2"

Find the distance between the points:-

color(blue)(B(8, 2) and C(4, 7)

Let us say the points are $B \left({x}_{1} , {y}_{1}\right) \mathmr{and} C \left({x}_{2} , {y}_{2}\right)$

Then, x_1 = 8; x_2=4, y_1 = 2 and y_2=7

Using the distance formula, we get

$\sqrt{{\left(4 - 8\right)}^{2} + {\left(7 - 2\right)}^{2}}$

$\Rightarrow \sqrt{16 + 25}$

$\Rightarrow \sqrt{41}$

$\approx 6.40312$

Hence, color(green)(bar(BC)~~6.40312

color(red)("Step 3"

Find the distance between the points:-

color(blue)(A(7, 5) and C(4, 7)

Let us say the points are $A \left({x}_{1} , {y}_{1}\right) \mathmr{and} C \left({x}_{2} , {y}_{2}\right)$

Then, x_1 = 7; x_2=4, y_1 = 5 and y_2=7

Using the distance formula, we get

$\sqrt{{\left(4 - 7\right)}^{2} + {\left(7 - 5\right)}^{2}}$

$\Rightarrow \sqrt{9 + 4}$

$\Rightarrow \sqrt{13}$

$\approx 3.60555$

Hence, color(green)(bar(AC)~~3.60555

color(red)("Step 4"

Perimeter of the triangle$= A B + B C + A C$

Perimeter $\approx 3.16228 + 6.40312 + 3.60555 \approx 13.171$

Hence,

Perimeter of the triangle ABC color(blue)(~~13.171 units.

View the image below for an illustration:
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