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

Mathematics

2 answers:

Svet_ta [12.7K]1 month ago

6 0

We are given the triangle

△ABC, with m∠A=60° and m∠C=45°, and AB=8.

To start, we will calculate all angles and sides.

Finding angle B:

The total of all angles in a triangle equals 180.

m∠A + m∠B + m∠C = 180.

Substituting the known values,

60° + m∠B + 45° = 180.

This gives us m∠B = 75°.

Calculating BC:

Using the law of sines,

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

We can substitute in the values.

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

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

$BC=9.798$

Finding AC:

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

Now we'll input the values.

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

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

$AC=10.928$

Calculating Perimeter:

$p=AB+BC+AC$

We substitute values here as well.

$p=10.928+8+9.798$

$p=28.726$

Calculating Area:

Using the area formula,

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

we can then insert values.

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

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

PIT_PIT [12.4K]1 month ago

6 0

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

Explanation:

Given:

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

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

The length of side AB is $AB = 8$

Calculation:

The total of all angles in 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 for 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 determined 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 is calculated as:

$\begin{aligned}{\text{Perimeter}}&= AB + BC + AC\\&= 8 + 9.80 + 10.93\\&= 28.73\\\end{aligned}$

The area of triangle ABC is calculated as:

$\begin{aligned}{\text{Area}}&=\frac{1}{2} \times AB \times AC \times \sin \left( A \right)\\&= \frac{1}{2}\times 8 \times 10.93 \times \sin {60^ \circ }\\&= 4\times 10.93 \times \frac{{\sqrt3 }}{2}\\&= 37.86\\\end{aligned}$

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

Learn more:

1. Understand more about the inverse of functions.

2. Explore equations of circles.

3. Learn about range and domain of functions

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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