Question

In the figure given below, a folding table is shown.

The legs of the table are represented by line segments AB and CD intersecting at O. Join AC and BD. Considering table top is a parallel to the ground and OB = x, OD = x + 3, OC = 3x + 19 and OA = 3x + 4, answer the following questions.

1. Prove that ΔOAC is similar to ΔOBD.    [1]
2. Prove that `(OA)/(AC) = (OB)/(BD)`.    [1]
3. 1. Observe the figure and find the value of x. Hence, find the length of OC.    [2]
      OR
   2. Observe the figure and find `(BD)/(AC)`.    [2]

Answer 1

Given OB = x, OD = x + 3, OC = 3x + 19 and OA = 3x + 4

and AC || BD

(i) ΔOAC ∼ ΔOBD

In ΔOAC and ΔOBD

∠DOB = ∠COA    ..[Vertically opposite angles]

As, AC is parallel to BD, AB and DC are transversal

∠BAC = ∠ABD

and ∠CDB = ∠DCA    ...[∵ Alternate interior angles]

ΔOAC ∼ ΔOBD    ...[by AA similarity criteria]

(ii) `(OA)/(AC) = (OB)/(BD)`

From part (i) we know that

ΔOAC ∼ ΔOBD

Corresponding sides of ΔOAC and ΔOBD must be proportional 

`(OA)/(OB) = (AC)/(BD)`

∴ `(OA)/(AC) = (OB)/(BD)` 

(iii)

(a) From part (i)

ΔOAC ∼ ΔOBD 

⇒ `(OA)/(OB) = (OC)/(OD)`

⇒ `(3x + 4)/x = (3x + 19)/(x + 3)`  ...[Given OA = 3x + 4, OB = x, OC = 3x + 19, OD = x + 3]

⇒ (3x + 4)(x + 3) = (3x + 19)x

⇒ 3x² + 9x + 4x + 12 = 3x² + 19x

⇒ 13x + 12 = 19x 

⇒ 19x − 13x = 12

⇒ 6x = 12

⇒ x = 2

Given, OC = 3x + 19

= 3 × (2) + 19

= 6 + 19

= 25

∴ OC = 25

OR 

(b) From part (ii), we have

`(OA)/(AC) = (OB)/(BD)`

`(BD)/(AC) = (OB)/(OA)`

`(BD)/(AC) = x/(3x + 4)`  ...[Given, OB = x and OA = 3x + 4]

On substituting x = 2, we get 

`(BD)/(AC) = 2/(3 xx 2 + 4)`

∴`(BD)/(AC) = 2/10`

∴`(BD)/(AC) = 1/5`