JEE Advanced · Mathematics · 17. Properties of Triangles

Let $A B C$ and $A B C^{\prime}$ be two non-congruent triangles with sides $A B=4$, $A C=A C^{\prime}=2 \sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

Verified Solution

Answer & Solution

Correct Answer

(D) 4

Step-by-step Solution

Detailed explanation

In $\triangle A B C$, by sine rule,
\[
\begin{aligned}
& \frac{a}{\sin A}=\frac{2 \sqrt{2}}{\sin 30^{\circ}}=\frac{4}{\sin C} \\
\Rightarrow \quad & C=45^{\circ}, C^{\prime}=135^{\circ} \\
\Rightarrow & A=180^{\circ}-\left(45^{\circ}+30^{\circ}\right)=105^{\circ}
\end{aligned}
\]

When $\quad C^{\prime}=135^{\circ}$, then
\[
\begin{aligned}
& \quad A=180^{\circ}-\left(135^{\circ}+30^{\circ}\right)=15^{\circ} \\
& \text { Area of } \triangle A B C=\frac{1}{2} A B \times A C \sin A \\
& =\frac{1}{2} \times 4 \times 2 \sqrt{2} \sin \left(105^{\circ}\right) \\
& =4 \sqrt{2} \times \frac{\sqrt{3}+1}{2 \sqrt{2}}=2(\sqrt{3}+1) \\
& \text { Area of } \triangle A B C^{\prime}=\frac{1}{2} A B \times A C \sin A \\
& =\frac{1}{2} \times 4 \times 2 \sqrt{2} \sin \left(15^{\circ}\right)=2(\sqrt{3}-1)
\end{aligned}
\]
Difference of areas of triangles
\[
=|2(\sqrt{3}+1)-2(\sqrt{3}-1)|=4
\]

\[
A D=2, D C=2
\]
Difference of areas of $\triangle A B C$ and $\triangle A B C^{\prime}$

\[
\begin{aligned}
& =\text { Area of } \triangle A C C^{\prime} \\
& =\frac{1}{2} A D \times C C^{\prime}=\frac{1}{2} \times 2 \times 4=4
\end{aligned}
\]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

Let $f, g : [- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of $f$ and $g$ at the points $- 1, 0$ and $2$ be as given in the following table:

$x = - 1$ $x = 0$ $x = 2$

$f (x)$ 3 6 0

$g (x)$ 0 1 -1

In each of the intervals $(- 1, 0)$ and $(0, 2)$ the function $(f - 3 g) "$ never vanishes. Then the correct statement(s) is(are)

JEE Advanced 2015 Medium

Column I		Column II
$(A)$	In a triangle $∆ X Y Z$ let $a, b$ and $c$ be the lengths of the angles $X, Y$ and $Z$ , respectively. If $2 (a^{2} - b^{2}) = c^{2}$ and $λ = \frac{\sin (X - Y)}{sinZ}$ then possible values of $n$ for which $\cos (n π λ) = 0$ is are	$(P)$	$1$
$(B)$	In a triangle $Δ X Y Z,$ let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$ , respectively. If $1 + \cos 2 x - 2 \cos 2 y$ $= 2 \sin x \sin y$ , then possible value(s) of $\frac{a}{b}$ is (are)	$(Q)$	$2$
$(C)$	In $R^{2},$ let $\sqrt{3} \hat{i} + \hat{j}, \hat{i} + \sqrt{3} \hat{j}$ and $β \hat{i} + (1 - β) \hat{j}$ be the position vectors of $X, Y$ and $Z$ with respect to the origin $O$ , respectively. If the distance of $Z$ from the bisector of the acute angle of $\vec{O X}$ with $\vec{O Y}$ is $\frac{3}{\sqrt{2}}$ , then possible value (s) of $\|β\|$ is (are)	$(R)$	$3$
$(D)$	Suppose that $F (α)$ denotes the area of the region bounded by $x = 0, x = 2, y^{2} = 4 x$ and $y = \|a x - 1\| +$ $\|α x - 2\| + α x,$ where $α \in \{0, 1\} .$ Then the value (s) of $F (α) + \frac{8}{3} \sqrt{2},$ when $α = 0$ and $α = 1$ , is (are)	$(S)$	$5$
		$(T)$	$6$

JEE Advanced 2015 Hard

Explore more questions on app

Let \(A B C\) and \(A B C^{\prime}\) be two non-congruent triangles with sides \(A B=4\), \(A C=A C^{\prime}=2 \sqrt{2}\) and angle \(B=30^{\circ}\). The absolute value of the difference between the areas of these triangles is

Answer & Solution

Step-by-step Solution

See the Complete Solution

	$x = - 1$	$x = 0$	$x = 2$
$f (x)$	3	6	0
$g (x)$	0	1	-1