Question

Given: △ABC, m∠A=60°,
m∠C=45°, AB=9
Find: Perimeter of △ABC,
Area of △ABC

Answer 1

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 $AB = 9$

Calculation:

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

$\begin{aligned}\angle A + \angle B + \angle C &= {180^ \circ }\\{60^ \circ } + \angle B + {45^ \circ }&= {180^ \circ }\\{105^ \circ } + \angle B &= {180^ \circ }\\\angle B &= {180^ \circ } - {105^ \circ }\\\angle B &= {75^ \circ }\\\end{aligned}$

The sine rule in triangle ABC can be expressed as,

$\begin{aligned}\frac{{BC}}{{\sin {{60}^\circ }}} &= \frac{9}{{\sin {{45}^\circ }}} \\BC&= \frac{9}{{\frac{1}{{\sqrt2 }}}} \times \frac{{\sqrt 3 }}{2}\\BC &= 11.02\\\end{aligned}$

The length of AC can be calculated as follows,

$\begin{aligned}\frac{{AB}}{{\sin {{45}^\circ }}}&= \frac{{AC}}{{\sin {{75}^\circ }}}\\\frac{9}{{\sin {{45}^\circ }}} \times \sin {75^\circ } &= AC\\12.30&= AC\\\end{aligned}$

The perimeter of triangle ABC can be obtained as follows,

$\begin{aligned}{\text{Perimeter}} &= AB + BC + AC \\ &= 9 + 11.02 + 12.30\\&= 32.32\\\end{aligned}$

The area of triangle ABC can be obtained as follows,

$\begin{aligned}{\text{Area}} &= \frac{1}{2} \times AB \times AC \times \sin \left( A \right)\\&= \frac{1}{2} \times 9 \times 12.30 \times \sin {60^\circ }\\&= 4.5 \times 12.30 \times \frac{{\sqrt3 }}{2}\\&= 47.93\\\end{aligned}$

The perimeter of triangle of ABC is $\boxed{32.32}$ and the area of triangle ABC is $\boxed{47.93}.$

Learn more:

1. Learn more about inverse of the function https://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 https://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.

Answer 2

Perimeter of △ABC given that m∠A=60°, m∠C=45° and AB=9 is 32.21

The perimeter of the triangle △ABC is expressed as:

Perimeter of △ABC = AB + BC + AC

Given the following measure

AB = 9

Get the measure of AC and BC using the sine rule;

m<B = 180 - ( m∠A +  m∠C)

m∠C = 180 - (60 + 45)

m∠C = 180 - 105

m∠C = 75 degrees

Get the length of BC

$\frac{BC}{sin \ m<A} =\frac{AB}{sin \ <C} \\\frac{BC}{sin 60} =\frac{9}{sin 45} \\BC = \frac{9sin60}{sin45}\\BC = \frac{9(0.8660)}{0.7071}\\BC=\frac{7.794}{0.7071}\\BC = 11.02$

Get the measure of AC

$\frac{AC}{sin \ m<B} =\frac{AB}{sin \ <C} \\\frac{AC}{sin 75} =\frac{9}{sin 45} \\AC = \frac{9sin75}{sin45}\\AC = \frac{9(0.9659)}{0.7071}\\AC=\frac{8.6933}{0.7071}\\AC = 12.29$

Get the perimeter of the triangle

Perimeter of △ABC = 12.29 + 9 + 11.02

Perimeter of △ABC = 32.31

Hence the perimeter of ABC is 32.31

Learn more here: https://brainly.com/question/23264228