Question

A Rhombus whose one side is 10 cm has one diagonal length is 12 cm. Find the measurement of other diagonals.​

Answer 1

» To Find :

The length of other diagonal .

» We Know :

Pythagoras theorem :

$\sf{\underline{\boxed{h^{2} = p^{2} + b^{2}}}}$

Where ,

• h = hypotenuse
• b = base
• p = height

» Concept :

We Know ,that the in rhombus the diagonals intersect each other equally and at 90°.

So the length AO will be Equal to length OC.

Hence , we get the length as :

$\sf{\underline{\boxed{AO = OC = 6 cm}}}$

And the given length is 10 cm .

According to the Diagram , if we look at the figure ADO ,we get a right-angled triangle having 90° at point O .

So by this information ,we can find the length OD , which is the corresponding height ,by using the Pythagoras theorem .

» Solution :

Given Information :

• Hypotenuse = 10 cm

• Base = 6 cm

Taken :

Let the Corresponding height be x

Formula :

$\sf{\underline{\boxed{h^{2} = p^{2} + b^{2}}}}$

By substituting the values in it ,we get :

$\sf{\Rightarrow 10^{2} = x^{2} + 6^{2}}$

Now, Subtracting 6² from both the sides ,we get :

$\sf{\Rightarrow 10^{2} - 6^{2} = x^{2} + 6^{2} - 6^{2}}$

$\sf{\Rightarrow 10^{2} - 6^{2} = x^{2}}$

By Square rooting on both the sides ,we get :

$\sf{\Rightarrow \sqrt{10^{2} - 6^{2}} = \sqrt{x^{2}}}$

$\sf{\Rightarrow \sqrt{10^{2} - 6^{2}} = x^{2}}$

$\sf{\Rightarrow \sqrt{100 - 36} = x}$

$\sf{\Rightarrow \sqrt{64} = x}$

$\sf{\Rightarrow 8 cm = x}$

Hence , the Corresponding height is 8 cm .

We Know, that the Diagonals intersect each other equally.

And the Corresponding height is the length DO , which is equal to OB ,so the other diagonal is twice of DO, i.e.

$\sf{\Rightarrow D_{2} = 2 \times DO}$

Substituting the value of DO ,in the above expression ,we get :

$\sf{\Rightarrow D_{2} = 2 \times 8}$

$\sf{\Rightarrow D_{2} = 16 cm}$

Hence , the other diagonal is 16 cm.

» Additional information :

• Volume of a Cylinder = πr²h

• Surface area of a Cylinder = 2πr(h + r)

• Curved-surface area of a Cylinder = 2πrh

• Area of triangle = ½ × base × height