SATHEE: Properties of Triangles 1 Question 1

Properties of Triangles 1 Question 1

1. The angles $A, B$ and $C$ of a $△ A B C$ are in $A P$ and $a : b = 1 : \sqrt{3}$ . If $c = 4 c m$ , then the area (in $s q c m$ ) of this triangle is

(2019 Main, 10 April II)

(a) $\frac{2}{\sqrt{3}}$

(b) $4 \sqrt{3}$

(d) $\frac{4}{\sqrt{3}}$

Show Answer

Answer:

Correct Answer: 1. (c)

Solution:

It is given that angles of a $△ A B C$ are in $A P$ .

So, $∠ A + ∠ B + ∠ C = 180^{\circ}$

$\Rightarrow ∠ B - d + ∠ B + ∠ B + d = 180^{\circ}$

[if $∠ A, ∠ B$ and $∠ C$ are in AP, then it taken as $∠ B - d$ , $∠ B, ∠ B + d$ respectively, where $d$ is common difference of AP]

$\begin{aligned} \Rightarrow 3 ∠ B = 180^{\circ} \Rightarrow ∠ B = 60^{\circ} \\ and \frac{a}{b} = \frac{1}{\sqrt{3}} \\ \Rightarrow \frac{\sin A}{\sin B} = \frac{1}{\sqrt{3}} \\ \Rightarrow \frac{\sin A}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \\ \Rightarrow \sin A = \frac{1}{2} \Rightarrow ∠ A = 30^{\circ} \\ [given. \\ So, ∠ C = 90^{\circ} \end{aligned}$

$∴$ From sine rule,

$\begin{aligned} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \\ \Rightarrow \frac{a}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{3}}{2}} = \frac{4}{1} \\ \Rightarrow a = 2 c m, b = 2 \sqrt{3} c m \\ ∴ Area of △ A B C = \frac{1}{2} a b \sin C = \frac{1}{2} \times 2 \times 2 \sqrt{3} \times 1 \\ = 2 \sqrt{3} s q \cdot c m \end{aligned}$

Properties of Triangles 1 Question 1

1. The angles $A, B$ and $C$ of a $△ A B C$ are in $A P$ and $a : b = 1 : \sqrt{3}$ . If $c = 4 c m$ , then the area (in $s q c m$ ) of this triangle is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Properties of Triangles 1 Question 1

1. The angles A,B and C of a △ABC are in AP and a:b=1:3. If c=4cm, then the area (in sqcm ) of this triangle is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

1. The angles $A, B$ and $C$ of a $△ A B C$ are in $A P$ and $a : b = 1 : \sqrt{3}$ . If $c = 4 c m$ , then the area (in $s q c m$ ) of this triangle is