Question

SAFE IS A RHOMBUS AS SHOW IN FIG 7 find x y z justify your answer( l want full answer with all steps) ​

Answer 1

☆ Question :

In the given figure , "SAFE" is a Rhombus , find the value of x , y and z.

☆ To Find :

The measures of x , y , and z.

☆ We Know :

Pythagoras theorem :

$\purple{\sf{\underline{\boxed{H^{2} = P^{2} + B^{2}}}}}$

Where ,

• H is the Hypotenuse

• P is the Height

• B is the Base

☆ Solution :

→ Concept :

We know the properties of Rhombus .i.e,

• Diagonals of a Rhombus intersect each other at 90° .

• Diagonals of a Rhombus intersect each other equally.

• All the sides of a Rhombus are equal.

So by using these properties we can find the value of x , y and z.

Value of x :

Given, Side OF = 9 cm

Using the Second Property of Rhombus .i.e,

"Diagonals of a Rhombus intersect each other equally".

So we get relation as , "OF = OS".

So the length of OS is also 9 cm.

Value of y :

Given, Side OA = 12 cm

Using the Second Property of Rhombus .i.e,

"Diagonals of a Rhombus intersect each other equally".

So we get relation as , "OA = OE".

So the length of OE is also 12 cm.

Value of z :

Using the First Property of Rhombus i.e,

"Diagonals of a Rhombus intersect each other at 90°".

Hence by this information , we know that a right-angled triangle will formed .

Taken Triangle is OAF

• Base(OF) = 9cm

• Height(OA) = 12 cm

So by using the Pythagoras theorem , we can find the other side i.e the hypotenuse , which will be the side of the Rhombus.

By Using the Pythagoras theorem and substituting the values in it , we get :

$\purple{\sf{H^{2} = P^{2} + B^{2}}} \\ \\ \\ \implies \sf{H^{2} = 12^{2} + 9^{2}}$

By Square Rooting on both the sides , we get :

$\implies \sqrt{\sf{H^{2}} = \sqrt{12^{2} + 9^{2}}} \\ \\ \\ \implies \sf{H = \sqrt{12^{2} + 9^{2}}} \\ \\ \\ \implies \sf{H = \sqrt{144 + 81}} \\ \\ \\ \implies \sf{H = \sqrt{225}} \\ \\ \\ \implies \sf{H = 15} \\ \\ \\ \therefore \purple{\sf{H = 15 cm}}$

Hence ,the hypotenuse of the triangle (OFA) is 15 cm .

But we know that the Hypotenuse of the Rhombus will be Equal to the Side of the Rhombus. So the side of the Rhombus is 15 cm.

Using the Third Property of Rhombus , i.e,

"All the sides of a Rhombus are equal".

Since , one side is 15 cm then the value of z is also 15 cm

☆ Answer :

• x = 9 cm
• y = 12 cm
• z = 15 cm

☆ Additional information :

• Area of a Rhombus = ½ × Product of its diagonals.

• Area of a Sector = lr/2

• Volume of a Cube = a³

• Surface area of Cuboid = 2(lh + lb + bh)