In a parallelogram, the sides are 24 cm and
10 cm and the diagonal is 26 cm. Find (i) its
area (ii) length of height corresponding to side
24 cm.
Answers
Answer:
i) The area of the parallelogram is 240 cm².
ii) The length of height of the parallelogram corresponding to the side 24 cm is 10 cm.
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
In figure, □ABCD is a parallelogram.
∴ AD = BC
AB = CD - - - ( 1 ) [ Opposite sides of parallelogram ]
Seg BD is the diagonal of the parallelogram.
Also,
AD = BC = 24 cm
AB = CD = 10 cm
BD = 26 cm
Now,
Draw diagonal AC joining the points A & C. - - - [ Construction ]
Now, we know that,
The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides of the parallelogram.
∴ ( AC )² + ( BD )² = ( AB )² + ( BC )² + ( CD )² + ( AD )²
⇒ AC² + BD² = AB² + BC² + AB² + BC² - - - [ From ( 1 ) ]
⇒ AC² + BD² = AB² + AB² + BC² + BC²
⇒ AC² + BD² = 2AB² + 2BC²
⇒ AC² + BD² = 2 ( AB² + BC² )
⇒ AC² + ( 26 )² = 2 [ ( 10 )² + ( 24 )² ] - - - [ Given ]
⇒ AC² + 676 = 2 ( 100 + 576 )
⇒ AC² + 676 = 2 * 676
⇒ AC² + 676 = 676 + 676
⇒ AC² = 676
⇒ AC = √676 - - - [ Taking square roots ]
⇒ AC = √( 26 * 26 )
⇒ AC = 26 cm
Now, in □ABCD,
AC = BD = 26 cm
∴ Diagonals of the parallelogram are congruent.
∴ □ABCD is a rectangle. - - - [ By definition ]
Now, in □ABCD,
m∠A = m∠B = m∠C = m∠D = 90° - - - [ Angle of a rectangle ]
∴ AB ⊥ BC
Now, we know that,
Area of parallelogram = Base * Height
⇒ A ( □ABCD ) = BC * AB
⇒ A ( □ABCD ) = 24 * 10
⇒ A ( □ABCD ) = 240 cm²
Now,
AB ⊥ BC
∴ Height of the parallelogram = AB
⇒ Height of the parallelogram = 10 cm
∴ The area of the parallelogram is 240 cm².
The length of height of the parallelogram corresponding to the side 24 cm is 10 cm.
