Question
ABCD is a quadrilateral.
$
\text { Is } A B+B C+C D+D A>A C+B D \text { ? }
$
$
\text { Is } A B+B C+C D+D A>A C+B D \text { ? }
$
✓
Answer
Yes, given $A B C D$ is a quadrilateral. Here, diagonal $A C$ divides the quadrilateral $A B C D$ into two $\triangle A B C$ and $\triangle A C D$.
In $\triangle A B C, \quad A B+B C>A C\quad\quad ...(i)$
In $\triangle A C D, \quad C D+D A>A C\quad\quad ...(ii)$
On adding Eqs. (i) and (ii), we get
$
A B+B C+C D+D A>2 A C \quad\quad ...(iii)$
Similarly, the diagonal $B D$ divides the quadrilateral $A B C D$ into two $\triangle B C D$ and $\triangle A B D$.
$
\begin{array}{ll}
\text { In } \triangle B C D, & B C+C D>B D \quad ...(iv)\\
\text { In } \triangle A B D, & A B+D A>B D \quad ...(v)
\end{array}
$
On adding Eqs. (iv) and (v), we get
$
(B C+C D+A B+D A)>2 B D \quad ...(vi)
$
Again, adding Eqs. (iii) and (vi), we get
$
\begin{array}{l}
A B+B C+C D+D A+B C+C D+A B+D A>2 A C+2 B D \\
\Rightarrow 2(A B+B C+C D+D A)>2(A C+B D) \\
\Rightarrow \quad A B+B C+D C+D A>A C+B D
\end{array}
$
In $\triangle A B C, \quad A B+B C>A C\quad\quad ...(i)$
In $\triangle A C D, \quad C D+D A>A C\quad\quad ...(ii)$
On adding Eqs. (i) and (ii), we get
$
A B+B C+C D+D A>2 A C \quad\quad ...(iii)$
Similarly, the diagonal $B D$ divides the quadrilateral $A B C D$ into two $\triangle B C D$ and $\triangle A B D$.
$
\begin{array}{ll}
\text { In } \triangle B C D, & B C+C D>B D \quad ...(iv)\\
\text { In } \triangle A B D, & A B+D A>B D \quad ...(v)
\end{array}
$
On adding Eqs. (iv) and (v), we get
$
(B C+C D+A B+D A)>2 B D \quad ...(vi)
$
Again, adding Eqs. (iii) and (vi), we get
$
\begin{array}{l}
A B+B C+C D+D A+B C+C D+A B+D A>2 A C+2 B D \\
\Rightarrow 2(A B+B C+C D+D A)>2(A C+B D) \\
\Rightarrow \quad A B+B C+D C+D A>A C+B D
\end{array}
$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Mr. Soni bought some bags of cement, each weighing $49.8\ kg$. If the total weight of all the bags is $1792.8\ kg$, how many bags did he buy?
→Find the value of the unknown interior angle $x$ in the figure:
→Find $(x + y) + (x - y)$, if$\text{x}=\frac{2}{3},\text{y}=\frac{3}{2}$
→Find the median of the given data if the mean is $4.5. 5, 7, 7, 8, x, 5, 4, 3, 1, 2$
→Among two supplementary angles, the measure of the larger angle is $36^\circ $ more than the measure of the smaller. Find their measures.
→Ravi delivers 2.5 kg, 3.2 kg, and 4.1 kg of vegetables to a store in the first three days. In 7 days, he delivers 20 kg of vegetables. What is the total quantity of vegetables delivered in the last four days?
One of the angles of a triangle is $100^\circ $ and the other two angles are equal. Find each of these equal angles.
→Simplify the following algebraic expressions by removing grouping symbols:
$20-\left[5 x y+3\left\{x^2-(x y-y)-(x-y)\right\}\right]$
→$20-\left[5 x y+3\left\{x^2-(x y-y)-(x-y)\right\}\right]$
If $x : y = 3 : 5$, find the ratio $3x + 4y : 8x + 5y.$
→If the area of a circle is 50.24 m2, find its circumference.