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nordsb [41]
3 years ago
12

The sum of the diagonals of a rhombus is 5√2.

Mathematics
1 answer:
Alex3 years ago
7 0
Greetings!
Let ABCD be a rhombus
AC + BD = 5√2 cm

Area of ABCD = Ar△ABD + Ar △BCD
= \frac{1}{2} \times BD \times AO + \frac{1}{2} \times BD \times OC
= \frac{1}{2} BD (AO + OC)
\frac{1}{2} BD \times AC

So, \frac{ BD \times AC}{2} = 4 \: cm {}^{2}
BD \times AC = 8
Now, AC + \: BD = 5 \sqrt{2}

Squaring both sides, we get
AC {}^{2} + BD {}^{2} + 2 AC.BD =50
AC {}^{2} + BD {}^{2} + 2 \times 8 = 50
AC {}^{2} + BD {}^{2} = 50 - 16
AC {}^{2} + BD {}^{2} = 34

In △AOB, we have
OA {}^{2} + OB {}^{2} =AB {}^{2}
( \frac{ AC }{2} ) {}^{2} + ( \frac{ BD}{2} ) {}^{2} = AB {}^{2}
\frac{ AC {}^{2} } {4} + \frac{ BD {}^{2} }{4} = AB {}^{2}
AC {}^{2} + BD {}^{2} = 4 AB {}^{2}
34 = 4 AB {}^{2}

Square rooting both sides
\sqrt{34} = 2 AB
Perimeter = 4 \: AB \\ = 2 \times 2 \: AB \\ = 2 \: \times \sqrt{34 } \\ = 2 \sqrt{34 \: } units.

Hope it helps!

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