Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Show that the Points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) Are the Vertices of a Rhombus. - Mathematics

Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus.

#### Solution

Let A(1,3,4) , B($-$1,6,10) , C($-$7,4,7) and D ($-$5,1,1) be the vertices of quadrilateral $\square ABCD$

$AB = \sqrt{\left( - 1 - 1 \right)^2 + \left( 6 - 3 \right)^2 + \left( 10 - 4 \right)^2}$
$= \sqrt{4 + 9 + 36}$
$= \sqrt{49}$
$= 7$
$BC = \sqrt{\left( - 7 + 1 \right)^2 + \left( 4 - 6 \right)^2 + \left( 7 - 10 \right)^2}$
$= \sqrt{36 + 4 + 9}$
$= \sqrt{49}$
$= 7$
$CD = \sqrt{\left( - 5 + 7 \right)^2 + \left( 1 - 4 \right)^2 + \left( 1 - 7 \right)^2}$
$= \sqrt{4 + 9 + 36}$
$= \sqrt{49}$
$= 7$
$DA = \sqrt{\left( 1 + 5 \right)^2 + \left( 3 - 1 \right)^2 + \left( 4 - 1 \right)^2}$
$= \sqrt{36 + 4 + 9}$
$= \sqrt{49}$
$= 7$
$\therefore AB = BC = CD = DA$

Hence, ABCD is a rhombus.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 28 Introduction to three dimensional coordinate geometry
Exercise 28.2 | Q 12 | Page 10