Question

BCDE is a rectangle, Z
/_OCD = 47° Find angles x, y and z. If BD = 5a - 3 and EC = a + 5. Find a, length of EC.
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Answer 1

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Answer 2

Given data :

• BCDE is a rectangle.
• ∠OCD = 47°
• BD = 5a - 3 and EC = a + 5

To find : Value of ‘a’ and EC and angles x, y and z.

Solution :

Here we know, from figure

→ ∠CDO or∠CDB = ∠ x

→ ∠EBD = ∠ y

→ ∠COB = ∠ z

According to figure,

→∠ECD or ∠OCD = 47°

Here, we know that the properties of the rectangle : The opposite sides are parallel and of equal length. Hence

→ ∠ECD = ∠CEB { Alternate angles }

→ ∠ECD = ∠CEB = 47°

Properties of the rectangle :

• The diagonals of a rectangle are of equal length.

Here from, property of rectangle,

→ BD = EC ......( 1 )

Properties of the rectangle :

The diagonals of a rectangle are bisect each other. Hence, ‘O’ is the mid point of diagonal BD and diagonal EC.

Now,

→ BD = BO + OD ......( 2 )

→ EC = EO + OC ......( 3 )

According to property of diagonal of rectangle,

→ BO = OD

→ EO = OC

Hence, eq. ( 2 ) become's

→ BD = OD + OD

→ BD = 2 OD ......( 4 )

and eq. ( 3 ) become's

→ EC = OC + OC

→ EC = 2 OC ......( 5 )

Now, from eq. ( 4 ) & eq. ( 5 )

∴ eq. ( 1 ) become's,

→ BD = EC

→ 2 OD = 2 OC

{ divide both side by 2 }

→ OD = OC {since, ‘O’ is the mid point of diagonal BD and diagonal EC}

→ ∠CDO = ∠OCD {angle opposite to equal sides are equal}

Similarly,

→ ∠CBD =∠BCO ......( 6 )

{angle opposite to equal sides are equal}

Hence,

→ ∠CDO = ∠OCD = 47°

∴ ∠CDO = 47°

∴ ∠x = 47°

Now, we know the opposite sides of rectangle are equal and parallel. Hence,

→ ∠CDO or∠CDB = ∠EBD {Alternate angle's}

→ ∠x = ∠y

∴ ∠x = ∠y = 47°

∴ ∠y = 47°

From the properties of the rectangle :

• Each vertex has angle equal to 90 degrees.
• All the interior angles of a rectangle are equal to 90 degrees.

Hence,

→∠EBD + ∠CBD = 90°

→ y + ∠CBD = 90°

→ 47° + ∠CBD = 90°

→ ∠CBD = 90° - 47°

→ ∠CBD = 43°

{ Note : ∠CBD or ∠CBO }

Now, from eq. ( 6 )

→ ∠CBD =∠BCO

∴ ∠BCO = 43°

Take Δ BOC from figure,

We know that, sum of the angles of the triangle are equal to 180°. Hence,

→∠CBO + ∠COB + ∠BCO = 180°

→ 43° + ∠COB + 43° = 180°

→ ∠COB + 86° = 180°

→ ∠COB = 180° - 86°

→ ∠COB = 94°

Here we know, ∠COB = z

∴ ∠ z = 94°

Now, from eq. ( 1 )

→ BD = EC

{from given}

→ 5a - 3 = a + 5

→ 5a - a = 5 + 3

→ 4a = 8

→ a = 8/4

→ a = 2

Hence,

→ EC = a + 5

→ EC = 2 + 5

→ EC = 7

Answer : Hence, value of ‘a’ is 2 and value of EC is 7 and also ∠x is 47°,∠y is 47° and ∠ z is 94°.