the complete question is
Heron’s formula: Area Use the triangle pictured to calculate the following measurements. Perimeter: p = units Semiperimeter: s = units Area: A ≈ square units. The triangle is 8 by 12 by 10
Part a) Calculate the Perimeter:
The perimeter of the triangle is the sum of its sides.
[tex] P = 8 + 12 + 10\\ P=30 units [/tex]
the answer Part a) is
The perimeter is equal to [tex] 30 units [/tex]
Part b) Calculate the Semi-perimeter:
The semiperimeter of a polygon is half its perimeter.
[tex] s = P / 2\\ s=30/2\\ s=15 units [/tex]
the answer part b) is
the semiperimeter is equal to [tex] 15 units [/tex]
Part c) Calculate the Area
we know that
You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that
It is called "Heron's Formula"
[tex] A = \sqrt{(s * (s-a) * (s-b) * (s-c))} [/tex]
where
s: is the semi-perimeter
a, b, c are the length sides.
[tex] s=15 units\\ a=8 units\\ b=12 units\\ c=10 units [/tex]
[tex] A = \sqrt{(15 * (15-8) * (15-12) * (15-10))} [/tex]
[tex] A = \sqrt{(15 * (7) * (3) * (5))} [/tex]
[tex] A = \sqrt{(15 * (105) )} [/tex]
[tex] A = \sqrt{(1,575 )} [/tex]
[tex] A= 39.7 units^{2} [/tex]
the answer Part c) is
[tex] A= 39.7 units^{2} [/tex]
The Perimeter, semi-perimeter, and area of a triangle with measurements 8 by 10 by 12 are 30 units, 15 units and 40 units square respectively.
Heron's formula is used in geometry to calculate the area of a triangle with respect to its sides. The formula is as follows:
[tex]\rm Area\:of\:triangle = \sqrt{s(s-a)(s-b)(s-c)}[/tex] , where s is the semi-perimeter of the triangle and a, b, and c are the length of its sides.
The perimeter of a triangle is the boundary of the triangle calculated as the sum of length of its sides. The formula therefore to calculate the perimeter of a triangle is:
[tex]\rm P = a + b + c[/tex] , where P is the perimeter of a triangle having sides a, b, and c.
Semi-perimeter is half the perimeter of a triangle. Therefore,
[tex]\rm s = \dfrac{P}{2}[/tex] , where s is the semi-perimeter.
Calculation of the perimeter of the triangle with sides 8,10, and 12 units:
[tex]\rm P = a + b + c\\\\\rm P = 8 + 10 + 12\\\\\rm P = 30 \:units[/tex]
Semi-perimeter of the triangle will be:
[tex]\rm s = \dfrac{P}{2}\\\\\rm s = \dfrac{30}{2}\\\\\rm s = 15\:units[/tex]
The area of the triangle will be:
[tex]\rm Area\:of\:triangle = \sqrt{s(s-a)(s-b)(s-c)}\\\\\rm Area\:of\:triangle = \sqrt{15(15-8)(15-10)(15-12)}\\\\\rm Area\:of\:triangle = \sqrt{15(7)(5)(3)}\\\\\rm Area\:of\:triangle = \sqrt{1575}\\\\\rm Area\:of\:triangle = 39.6862 \:units^2\\\\\rm Area\:of\:triangle =40 \:units^2\:(rounded)\\[/tex]
Therefore the perimeter is 30 units, semi-perimeter is 15 units and area is 40 units square.
Learn more about Heron's formula here:
https://brainly.com/question/22391198
