Question

The parallel sides of a trapezium are 12 cm and 17 cm. Its non-parallel sides are equal, each
of length 6.5 cm. Considering the trapezium as the sum of a parallelogram and an isosceles
triangle, find its area.
10 cm​

Answer 1

Answer:

The area of the trapezium is 87 cm².

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, □ABCD is a trapezium.

AD ∥ BC

AB ∦ CD

We have given that,

AD = 12 cm

BC = 17 cm

AB = CD = 6.5 cm

Draw seg DE such that DE ∥ AB. - - - [ Construction ]

Now, in □ABED,

AD ∥ BC - - - [ Given ]

∴ AD ∥ BE - - - [ B - E - C ]

Also,

AB ∥ DE - - - [ Construction ]

∴ □ABED is a parallelogram. - - - [ By definition ]

∴ AB = DE = 6.5 cm - - - [ Opposite sides of parallelogram ]

AD = BE = 12 cm - - - [ Opposite sides of parallelogram ]

Now,

BC = 17 cm - - - [ Given ]

⇒ BE + CE = 17 cm - - - [ B - E - C ]

⇒ 12 + CE = 17 cm

⇒ CE = 17 - 12 cm

⇒ CE = 5 cm

Now, in △DEC,

DE = 6.5 cm

DC = 6.5 cm

CE = 5 cm

∴ △DEC is an isosceles triangle.

Draw DM ⊥ BC i. e. EC. - - - [ Construction ]

Now, we know that,

An altitude drawn in an isosceles triangle from the common vertex of congruent sides to the third side bisects the third side.

∴ In △DEC,

EM = CM = ½ * CE

⇒ CM = ½ * 5

⇒ CM = 5 ÷ 2

⇒ CM = 2.5 cm

Now, in △DMC, m∠DMC = 90°,

∴ ( DC )² = ( DM )² + ( CM )² - - - [ Pythagoras theorem ]

⇒ DC² = DM² + CM²

⇒ DM² = DC² - CM²

⇒ DM² = ( 6.5 )² - ( 2.5 )²

⇒ DM² = ( 6.5 + 2.5 ) ( 6.5 - 2.5 ) - - - [ ∵ a² - b² = ( a + b ) ( a - b ) ]

⇒ DM² = 9 * 4

⇒ DM = √( 9 * 4 ) - - - [ Taking square roots ]

⇒ DM = √( 3 * 3 * 2 * 2 )

⇒ DM = 3 * 2

⇒ DM = 6 cm

Now, we know that,

Area of trapezium = ( Sum of parallel sides * Height ) / 2

⇒ A ( □ABCD ) = [ ( AD + BC ) * DM ] / 2

⇒ A ( □ABCD ) = [ ( 12 + 17 ) * 6 ] / 2

⇒ A ( □ABCD ) = 12 + 17 * 6 / 2

⇒ A ( □ABCD ) = 29 * 6 ÷ 2

⇒ A ( □ABCD ) = 29 * 3

⇒ A ( □ABCD ) = 87 cm²

∴ The area of the trapezium is 87 cm².

Answer 2

Answer:

The parallel sides of a trapezium are 12 cm and 17 cm. Its non-parallel sides are equal, each

of length 6.5 cm. Considering the trapezium as the sum of a parallelogram and an isosceles

triangle, find its area.

10 cm