Question

Find the perimeter of the triangles whose vertices have the following coordinates\[\left( {3,10} \right)\] \[\left( {5,2} \right)\]and\[\left( {14,12} \right)\].

Answer 1

Hint- Perimeter is the total length of the outer boundary of a closed region, so the perimeter of the triangle is the total sum of its all three sides. The perimeter of the triangle is \[P = a + b + c\]where \[a,b,c\]is the length of the sides of the triangle, with all sides having the same units. If the length of the side of the triangle is not given then, find the side’s length using the vertices of the triangle, which are given in this question. If the vertices of the triangle are given, we need to find the distance between them, which will be their side’s length. We can find the distance between the vertices \[\left( {{x_1},{y_1}} \right)\]and \[\left( {{x_2},{y_2}} \right)\] , by using formula \[d = \sqrt {{{\left( {{x_2} - {x_{_1}}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]hence, \[d\]becomes the side lengths of the triangle. Using this formula, find other sides of the triangle, and these side’s length will be used to find the area and perimeter of the triangle.

Complete step by step solution:
Let the vertices of the triangle whose perimeter is to be found by A, B, C where
\[A = \left( {3,10} \right)\]
\[B = \left( {5,2} \right)\]
\[C = \left( {14,12} \right)\]
Hence the distance between the pair of vertices will be
\[AB = {L_1} = \sqrt {{{\left( {5 - 3} \right)}^2} + {{\left( {2 - 10} \right)}^2}} = \sqrt {{2^2} + {{\left( { - 8} \right)}^2}} = \sqrt {4 + 64} = \sqrt {68} = 8.25\]
\[BC = {L_2} = \sqrt {{{\left( {14 - 5} \right)}^2} + {{\left( {12 - 2} \right)}^2}} = \sqrt {{9^2} + {{10}^2}} = \sqrt {81 + 100} = \sqrt {181} = 13.45\]
\[CA = {L_3} = \sqrt {{{\left( {3 - 14} \right)}^2} + {{\left( {10 - 12} \right)}^2}} = \sqrt {{{\left( { - 11} \right)}^2} + {{\left( { - 2} \right)}^2}} = \sqrt {121 + 4} = \sqrt {125} = 11.18\]
Now we know the length of the sides of triangle so we can find the perimeter of the triangle where the perimeter of the triangle is
\[
  P = {L_1} + {L_2} + {L_3} \\
   = 8.25 + 13.45 + 11.18 \\
   = 32.88 \\
 \]
Hence, the perimeter of the triangle whose vertices are \[\left( {3,10} \right)\] \[\left( {5,2} \right)\]and\[\left( {14,12} \right)\]is 32.88 units.
Note: To find the perimeter or area of a triangle, first calculate the length of the sides of the triangle from the vertices of the triangle. The area of the triangle using its side’s length can be determined by using Heron’s Formula.