The perimeter of triangle of ABC is ![\boxed{28.73}.](https://tex.z-dn.net/?f=%5Cboxed%7B28.73%7D.)
Further explanation:
Given:
The measure of angle A is ![\angle A = {60^ \circ }.](https://tex.z-dn.net/?f=%5Cangle%20A%20%3D%20%7B60%5E%20%5Ccirc%20%7D.)
The measure of angle C is ![\angle C = {45^ \circ }.](https://tex.z-dn.net/?f=%5Cangle%20C%20%3D%20%7B45%5E%20%5Ccirc%20%7D.)
The length of side AB is ![AB = 8](https://tex.z-dn.net/?f=AB%20%3D%208)
Calculation:
The sum of all angles of a triangle is ![{180^ \circ }.](https://tex.z-dn.net/?f=%7B180%5E%20%5Ccirc%20%7D.)
![\begin{aligned}\angle A + \angle B + \angle C&={180^ \circ }\\{60^ \circ } + \angle B + {45^ \circ }&= {180^ \circ }\\{105^ \circ }+\angle B&= {180^ \circ }\\\angleB&= {180^ \circ } - {105^ \circ }\\\angleB&= {75^ \circ }\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cangle%20A%20%2B%20%5Cangle%20B%20%2B%20%5Cangle%20C%26%3D%7B180%5E%20%5Ccirc%20%7D%5C%5C%7B60%5E%20%5Ccirc%20%7D%20%2B%20%5Cangle%20B%20%2B%20%7B45%5E%20%5Ccirc%20%7D%26%3D%20%7B180%5E%20%5Ccirc%20%7D%5C%5C%7B105%5E%20%5Ccirc%20%7D%2B%5Cangle%20B%26%3D%20%7B180%5E%20%5Ccirc%20%7D%5C%5C%5CangleB%26%3D%20%7B180%5E%20%5Ccirc%20%7D%20-%20%7B105%5E%20%5Ccirc%20%7D%5C%5C%5CangleB%26%3D%20%7B75%5E%20%5Ccirc%20%7D%5C%5C%5Cend%7Baligned%7D)
The sine rule in triangle ABC can be expressed as,
![\begin{aligned}\frac{{BC}}{{\sin {{60}^ \circ }}}&=\frac{8}{{\sin {{45}^ \circ }}}\\BC&= \frac{8}{{\frac{1}{{\sqrt2 }}}} \times \frac{{\sqrt 3 }}{2}\\BC &= 9.80\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cfrac%7B%7BBC%7D%7D%7B%7B%5Csin%20%7B%7B60%7D%5E%20%5Ccirc%20%7D%7D%7D%26%3D%5Cfrac%7B8%7D%7B%7B%5Csin%20%7B%7B45%7D%5E%20%5Ccirc%20%7D%7D%7D%5C%5CBC%26%3D%20%5Cfrac%7B8%7D%7B%7B%5Cfrac%7B1%7D%7B%7B%5Csqrt2%20%7D%7D%7D%7D%20%5Ctimes%20%5Cfrac%7B%7B%5Csqrt%203%20%7D%7D%7B2%7D%5C%5CBC%20%26%3D%209.80%5C%5C%5Cend%7Baligned%7D)
The length of AC can be calculated as follows,
![\begin{aligned}\frac{{AB}}{{\sin {{45}^ \circ }}}&=\frac{{AC}}{{\sin {{75}^ \circ }}}\\\frac{8}{{\sin {{45}^ \circ }}}\times \sin {75^ \circ }&= AC\\10.93& = AC\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cfrac%7B%7BAB%7D%7D%7B%7B%5Csin%20%7B%7B45%7D%5E%20%5Ccirc%20%7D%7D%7D%26%3D%5Cfrac%7B%7BAC%7D%7D%7B%7B%5Csin%20%7B%7B75%7D%5E%20%5Ccirc%20%7D%7D%7D%5C%5C%5Cfrac%7B8%7D%7B%7B%5Csin%20%7B%7B45%7D%5E%20%5Ccirc%20%7D%7D%7D%5Ctimes%20%5Csin%20%7B75%5E%20%5Ccirc%20%7D%26%3D%20AC%5C%5C10.93%26%20%3D%20AC%5C%5C%5Cend%7Baligned%7D)
The perimeter of triangle ABC can be obtained as follows,
![\begin{aligned}{\text{Perimeter}}&= AB + BC + AC\\&= 8 + 9.80 + 10.93\\&= 28.73\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B%5Ctext%7BPerimeter%7D%7D%26%3D%20AB%20%2B%20BC%20%2B%20AC%5C%5C%26%3D%208%20%2B%209.80%20%2B%2010.93%5C%5C%26%3D%2028.73%5C%5C%5Cend%7Baligned%7D)
The area of triangle ABC can be obtained as follows,
![\begin{aligned}{\text{Area}}&=\frac{1}{2} \times AB \times AC \times \sin \left( A \right)\\&= \frac{1}{2}\times 8 \times 10.93 \times \sin {60^ \circ }\\&= 4\times 10.93 \times \frac{{\sqrt3 }}{2}\\&= 37.86\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B%5Ctext%7BArea%7D%7D%26%3D%5Cfrac%7B1%7D%7B2%7D%20%5Ctimes%20AB%20%5Ctimes%20AC%20%5Ctimes%20%5Csin%20%5Cleft%28%20A%20%5Cright%29%5C%5C%26%3D%20%5Cfrac%7B1%7D%7B2%7D%5Ctimes%208%20%5Ctimes%2010.93%20%5Ctimes%20%5Csin%20%7B60%5E%20%5Ccirc%20%7D%5C%5C%26%3D%204%5Ctimes%2010.93%20%5Ctimes%20%5Cfrac%7B%7B%5Csqrt3%20%7D%7D%7B2%7D%5C%5C%26%3D%2037.86%5C%5C%5Cend%7Baligned%7D)
The perimeter of triangle of ABC is
and the area of triangle ABC is ![\boxed{37.86}.](https://tex.z-dn.net/?f=%5Cboxed%7B37.86%7D.)
Learn more:
1. Learn more about inverse of the function brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497
Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Triangles
Keywords: angles, ABC, angle A=60 degree, perimeter, area of triangle, triangle ABC.