Page 76 - Math_Genius_V1.0_C8_Flipbook

E:\Working\Focus_Learning\Math_Genius-8\Open_Files\03_Chapter_3\Chapter_3
\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Example 25: The length of a rectangle is 8 cm and each of its diagonals measures 10 cm. Find its
breadth.

Solution: Let ABCD be the given rectangle in which length AB = 8 cm and D C
diagonal AC = 10 cm.
Since, each angle of a rectangle is a right angle, we have ∠ABC = 90°. 10 cm

From right ∆ABC, we have

AB² + BC² = AC² [Pythagoras Theorem]
BC² = (AC² – AB²) = {(10)² – (8)²} A 8 cm B

= 100 – 64 = 36 = 6 2
\ BC = 6 cm

Thus, the breadth of the rectangle is 6 cm.

Example 26: The adjacent sides of a rectangle are in ratio 5 : 12. If the perimeter of the rectangle
is 34 cm, find the diagonal of the rectangle.
Solution: Given, the adjacent sides are in ratio 5 : 12.
A B
Let the adjacent sides of the rectangle be 5x and 12x.
Then, 5x + 12x + 5x + 12x = 34 5 cm

34x = 34 D 12 cm C

⇒ x = 1
Thus, the sides of the rectangle are 5 cm and 12 cm.

Now, by Pythagoras Theorem,
2
2
2
2
2
BD = BC + DC = 5 + 12 = 25 + 144 = 169 = 13 2
\ The length of the diagonal = 13 cm
Square
A rectangle with all the four equal sides is called a square. In the adjoining A B
figure, ABCD is a square where, AB = BC = CD = AD.

Properties of a Square E

• All four sides are equal, i.e., AB = BC = CD = DA.

• Opposite sides are parallel, i.e., AB || CD and AD || BC
D C
• Diagonals are of equal length, i.e., AC = BD

• Diagonals are perpendicular bisector to each other.
• Diagonals bisects the angles, DB is the bisector of ∠ADC and ∠ABC. Similarly, AC is the bisector
of ∠BCD and ∠BAD.

• All the four angles are of 90°.

• Each diagonal divides the square into two congruent triangles.

Mathematics-8 74