Question

In the adjoining figure, ABCD is a rectangle. Find x, y and z.

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Answer 1

Question:

In the adjoining figure, ABCD is a rectangle. Find x, y and z.

Required answer:

The values of x, y and z are 42°, 48° and 48° respectively.

Explaination:

Given that :

⟼ ABCD is a rectangle

This means that :

⟼ AC = BD . . . . . (eqⁿ. 1)

Diagonals in a rectangle are equal in length.

⟼ AO = OC . . . . . (eqⁿ. 2)

⟼ DO = OB . . . . . (eqⁿ. 3)

Diagonals in a rectangle bisect each other.

We know that :

⟼ AC = AO + OC

⟼ BD = DO + OB

From (eqⁿ. 1), we get :

⟼ AO + OC = DO + OB

From (eqⁿ. 2, 3), we get :

⟼ 2AO = 2OB

⟼ AO = OB . . . . . (eqⁿ. 4)

From, (eqⁿ. 4), we can say that :

⟼ ∆AOB is an isosceles triangle

In an isosceles triangle ∆AOB, with sides AO = OB :

⟼ ∠OAB = ∠OBA

⟼ ∠OAB = ∠OBA = x

Angles opposite to equal sides are equal.

Using angle sum property of a triangle, we'll calculate the value of x :

⟼ ∠AOB + ∠OAB + ∠OBA = 180°

⟼ 96° + x + x = 180°

⟼ 96° + 2x = 180°

⟼ 2x = 180° – 96°

⟼ 2x = 84°

⟼ x = 84/2

⟼ x = 42°

⟼ ∠OAB = ∠OBA = x = 42°

Now,

⟼ ∠COB + ∠AOB = 180°

AOC is a straight line and these angles form a linear pair.

Let's find ∠COB :

⟼ ∠COB + 96° = 180°

⟼ ∠COB = 180° – 96°

⟼ ∠COB = 84°

Again from (eqⁿ. 1), we get :

⟼ AO + OC = DO + OB

From (eqⁿ. 2, 3), we get :

⟼ 2OC = 2OB

⟼ OC = OB . . . . . (eqⁿ. 5)

From, (eqⁿ. 5), we can say that :

⟼ ∆COB is an isosceles triangle

In an isosceles triangle ∆COB, with sides CO = OB :

⟼ ∠OBC = ∠OCB = y

Angles opposite to equal sides are equal.

Let's find the value of y using the angle sum property of rectangle :

⟼ ∠OBC + ∠OCB + ∠COB = 180°

⟼ y + y + 84° = 180°

⟼ 2y + 84° = 180°

⟼ 2y = 180° – 84°

⟼ 2y = 96°

⟼ y = 96/2

⟼ y = 48°

⟼ ∠OBC = ∠OCB = y = 48°

Here :

⟼ ∠COB = ∠AOD

⟼ ∠AOD = 84°

Vertically opposite angles are equal.

Finally, from (eqⁿ. 1) :

⟼ AO + OC = DO + OB

From, (eqⁿ. 2, 3), we get :

⟼ 2AO = 2DO

⟼ AO = DO . . . . . (eqⁿ. 6)

From, (eqⁿ. 6), we get :

⟼ ∆AOD is an isosceles triangle

In an isosceles triangle ∆AOD, with sides AO = DO :

⟼ ∠ODA = ∠OAD = z

Angles opposite to equal sides are equal.

Finding the value of z using the angle sum property of a triangle :

⟼ ∠ODA + ∠OAD + ∠AOD = 180°

⟼ z + z + 84° = 180°

⟼ 2z = 180° – 84°

⟼ 2z = 96°

⟼ z = 96/2

⟼ z = 48°

⟼ ∠ODA = ∠OAD = z = 48°