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Quadrilaterals — Maths STD 9 — Question

CBSE BoardEnglish MediumSTD 9MathsQuadrilaterals1 Mark
MCQ
In a trapezium ABCD, if $A B \| C D$, then $A C^2+B D^2=$
Image
  • A
    $B C^2+A D^2+2 B C \times A D$
  • B
    $A B^2+C D^2+2 A B \times C D$
  • C
    $A B^2+C D^2+2 A D \times B C$
  • ✓
    $B C^2+A D^2+2 A B \times C D$
✓

Answer

Correct option: D.
$B C^2+A D^2+2 A B \times C D$
(d) $B C^2+A D^2+2 A B \times C D$
In $\triangle A B C, \angle B$ is acute angle.
$\therefore \quad A C^2=A B^2+B C^2-2 A B \times B F \quad$...(i)
In $\triangle A B D, \angle A$ is acute angle.
$\therefore \quad B D^2=A B^2+A D^2-2 A B \times A E \quad$...(ii)
Adding (i) and (ii), we get
$\begin{aligned}A C^2+B D^2 & =B C^2+A D^2+2 A B^2-2 A B \times B F-2 A B \times A E \\
& =B C^2+A D^2+2 A B(A B-B F-A E) \\
& =B C^2+A D^2+2 A B \times E F=B C^2+A D^2+2 A B \times C D\end{aligned}$

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