[tex]\bold{\huge{\purple{\underline{ Solutions}}}}[/tex]
Answer 5 :-
We have,
[tex]\sf{ HE = 2x cm \: and \: OR = ( x + 5)cm}[/tex]
We know that,
- Parallelogram has opposite sides parallel and equal.
From above property,
[tex]\sf{ HE = OR }[/tex]
[tex]\sf{ 2x = x + 5}[/tex]
[tex]\sf{ 2x - x = 5}[/tex]
[tex]\sf{ x = 5}[/tex]
Thus, The value of x is 5cm
Therefore,
The value of HE
[tex]\sf{ = 2x}[/tex]
[tex]\sf{ = 2× 5}[/tex]
[tex]\sf{\red{ = 10 cm}}[/tex]
The measurement of HE is 10 cm
Hence, Option A is correct
Answer 6 :-
We have,
[tex]\sf{m}{\sf{\angle{HER = 5y - 26° \: and\: m }}}{\sf{\angle{ ROH = 2y + 40°}}}[/tex]
- In parallelograms, Opposite angles are equal
From above property,
[tex]\sf{m}{\sf{\angle{HER = }}}{\sf{\angle{ ROH}}}[/tex]
[tex]\sf{ 5y - 26° = 2y + 40°}[/tex]
[tex]\sf{ 5y - 2y = 40 + 26}[/tex]
[tex]\sf{ 3y = 40 + 26}[/tex]
[tex]\sf{ y = 66/3}[/tex]
[tex]\sf{ y = 22°}[/tex]
Thus, The value of y is 22°
Therefore,
The measurement of Angle HER
[tex]\sf{ = 5 × 22 - 26 }[/tex]
[tex]\sf{ = 110 - 26 }[/tex]
[tex]\sf{\red{= 84° }}[/tex]
The measurement of angle HER is 84°
Hence option D is correct
Answer 7 :-
We have,
[tex]\sf{ HZ = 4a - 5 \:and \:RZ = 3a + 5 }[/tex]
- The diagonals of parallelogram are equal in length
- Here, RZ is the bisector of diagonal HR
[tex]\sf{ 4a - 5 = 3a + 5 }[/tex]
[tex]\sf{ 4a - 3a = 5 + 5}[/tex]
[tex]\sf{ a = 10}[/tex]
Thus, The value of a is 10
Therefore,
The measurement of HZ
[tex]\sf{ = 4 × 10 - 5 }[/tex]
[tex]\sf{ = 40 - 5 }[/tex]
[tex]\sf{\red{= 35 \: cm }}[/tex]
The measurement of HZ is 35 cm
Hence, Option C is correct
Answer 8 :-
We have ,
One diagonals = 5x - 47
Another diagonal = 2x + 34
- In isosceles trapezoid, Diagonals are equal in length.
From above property,
[tex]\sf{ 5x - 47 = 2x + 34 }[/tex]
[tex]\sf{ 5x - 2x = 34 + 47}[/tex]
[tex]\sf{ 3x = 81 }[/tex]
[tex]\sf{ x = 81/3 }[/tex]
[tex]\sf{ x = 27}[/tex]
Thus, The value of x is 27
Hence, Option A is correct
Answer 9 :-
We have,
The sides of parallelogram GAME are (7x - 1 ) , ( 5x + 10 ) , (6x) , ( 7x - 2 )
- In parallelogram, Opposite sides are equal
From above property,
[tex]\sf{ 7x - 1 = 6x }[/tex]
[tex]\sf{ 7x - 6x = 1 }[/tex]
[tex]\sf{ x = 1 }[/tex]
[tex]\sf{ 5x + 10 = 7x - 2}[/tex]
[tex]\sf{ 7x - 5x = 10 + 2 }[/tex]
[tex]\sf{ 2x = 12}[/tex]
[tex]\sf{ x = 12/2 }[/tex]
[tex]\sf{ x = 6 }[/tex]
Thus, measurements of parallelogram GAME in inches is 1 and 6 .
Hence, Option B is correct
Answer 10 :-
We have,
[tex]\sf{ ES = 18 cm}[/tex]
- The diagonals of rectangle are equal in length.
- Here, NP = OE
- PS and NS act as a bisector in OE and OS and ES act as a bisector in NP
From above we can say that,
[tex]\sf{ OE = OS + ES }[/tex]
[tex]\sf{ OE = 2ES ( OS = ES bisector}[/tex]
[tex]\sf{ OE = 2 × 18 }[/tex]
[tex]\sf{ OE = 36cm }[/tex]
Thus, The value of OE is 38 cm
Therefore,
[tex]\sf{ OE = NP }[/tex]
[ Diagonals of the given rectangle ]
[tex]\sf{\red{NP = 36 cm }}[/tex]
Thus , The value of NP is 36 cm
Hence, Option C is correct
