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Dima020 [189]
3 years ago
5

Given: △ABC, m∠A=60° m∠C=45°, AB=8 Find: Perimeter of △ABC, Area of △ABC

Mathematics
2 answers:
jarptica [38.1K]3 years ago
6 0

We are given

△ABC, m∠A=60° m∠C=45°, AB=8

Firstly, we will find all angles and sides

Calculation of angle B:

we know that sum of all angles is 180

m∠A+ m∠B+m∠C=180

we can plug values

60°+ m∠B+45°=180

m∠B=75°

Calculation of BC:

we can use law of sines

\frac{AB}{sin(C)}=\frac{BC}{sin(A)}

now, we can plug values

\frac{8}{sin(45)}=\frac{BC}{sin(60)}

BC=\frac{8}{sin(45)} \times sin(60)

BC=9.798

Calculation of AC:

\frac{AB}{sin(C)}=\frac{AC}{sin(B)}

now, we can plug values

\frac{8}{sin(45)}=\frac{AC}{sin(75)}

AC=\frac{8}{sin(45)} \times sin(75)

AC=10.928

Perimeter:

p=AB+BC+AC

we can plug values

p=10.928+8+9.798

p=28.726

Area:

we can use formula

A=\frac{1}{2}AB \times AC \times sin(A)

now, we can plug values

A=\frac{1}{2}8 \times 10.928 \times sin(60)

A=37.85570...............Answer

igomit [66]3 years ago
6 0

The perimeter of triangle of ABC is \boxed{28.73}.

Further explanation:

Given:

The measure of angle A is \angle A = {60^ \circ }.

The measure of angle C is \angle C = {45^ \circ }.

The length of side AB is AB = 8

Calculation:

The sum of all angles of a triangle is {180^ \circ }.

\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}

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}

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}

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}

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}

The perimeter of triangle of ABC is \boxed{28.73} and the area of triangle ABC is \boxed{37.86}.

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.

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