Quadrilaterals - Exercise 13.4 | Extra Documents & Tests for Class 9 PDF Download

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Q u e s t i o n : 2 4
In a ?ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are
7 cm, 8 cm and 9 cm, respectively, find the perimeter of ?DEF.
S o l u t i o n :
 is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:
Page 2


     
            
     
          
    
  
     
    
      
        
 
  
 
Q u e s t i o n : 2 4
In a ?ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are
7 cm, 8 cm and 9 cm, respectively, find the perimeter of ?DEF.
S o l u t i o n :
 is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:
Also,  , and .
We need to find the perimeter of 
In , E and F are the mid-points of CA and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
And
Perimeter of 
Hence, the perimeter of  is .
Q u e s t i o n : 2 5
In a triangle ?ABC, ?A = 50°, ?B = 60° and C = ?70°. Find the measures of the angles of the triangle formed by
joining the mid-points of the sides of this triangle.
S o l u t i o n :
It is given that D, E and F be the mid-points of BC , CA and AB respectively.
Then,
, and .
Page 3


     
            
     
          
    
  
     
    
      
        
 
  
 
Q u e s t i o n : 2 4
In a ?ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are
7 cm, 8 cm and 9 cm, respectively, find the perimeter of ?DEF.
S o l u t i o n :
 is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:
Also,  , and .
We need to find the perimeter of 
In , E and F are the mid-points of CA and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
And
Perimeter of 
Hence, the perimeter of  is .
Q u e s t i o n : 2 5
In a triangle ?ABC, ?A = 50°, ?B = 60° and C = ?70°. Find the measures of the angles of the triangle formed by
joining the mid-points of the sides of this triangle.
S o l u t i o n :
It is given that D, E and F be the mid-points of BC , CA and AB respectively.
Then,
, and .
Now,  and transversal CB and CA intersect them at D and E respectively.
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
] and 
[ Given
]
Now BC is a straight line.
Similarly, 
and 
Hence the measure of angles are , and .
Q u e s t i o n : 2 6
In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and
AB = 30 cm, find the perimeter of the quadrilateral ARPQ.
S o l u t i o n :
It is given that P, Q and R are the mid-points of BC, CA and AB respectively.
Page 4


     
            
     
          
    
  
     
    
      
        
 
  
 
Q u e s t i o n : 2 4
In a ?ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are
7 cm, 8 cm and 9 cm, respectively, find the perimeter of ?DEF.
S o l u t i o n :
 is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:
Also,  , and .
We need to find the perimeter of 
In , E and F are the mid-points of CA and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
And
Perimeter of 
Hence, the perimeter of  is .
Q u e s t i o n : 2 5
In a triangle ?ABC, ?A = 50°, ?B = 60° and C = ?70°. Find the measures of the angles of the triangle formed by
joining the mid-points of the sides of this triangle.
S o l u t i o n :
It is given that D, E and F be the mid-points of BC , CA and AB respectively.
Then,
, and .
Now,  and transversal CB and CA intersect them at D and E respectively.
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
] and 
[ Given
]
Now BC is a straight line.
Similarly, 
and 
Hence the measure of angles are , and .
Q u e s t i o n : 2 6
In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and
AB = 30 cm, find the perimeter of the quadrilateral ARPQ.
S o l u t i o n :
It is given that P, Q and R are the mid-points of BC, CA and AB respectively.
Also, we have , and
We need to find the perimeter of quadrilateral ARPQ
In , P and R are the mid-points of CB and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a traingle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
We have Q and R as the mid points of AC and AB respectively.
Therefore,
And
Perimeter of 
Hence, the perimeter of quadrilateral ARPQ is .
Page 5


     
            
     
          
    
  
     
    
      
        
 
  
 
Q u e s t i o n : 2 4
In a ?ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are
7 cm, 8 cm and 9 cm, respectively, find the perimeter of ?DEF.
S o l u t i o n :
 is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:
Also,  , and .
We need to find the perimeter of 
In , E and F are the mid-points of CA and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
And
Perimeter of 
Hence, the perimeter of  is .
Q u e s t i o n : 2 5
In a triangle ?ABC, ?A = 50°, ?B = 60° and C = ?70°. Find the measures of the angles of the triangle formed by
joining the mid-points of the sides of this triangle.
S o l u t i o n :
It is given that D, E and F be the mid-points of BC , CA and AB respectively.
Then,
, and .
Now,  and transversal CB and CA intersect them at D and E respectively.
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
]
and 
 [ Given
]
Similarly, 
Therefore,
 [ Given
] and 
[ Given
]
Now BC is a straight line.
Similarly, 
and 
Hence the measure of angles are , and .
Q u e s t i o n : 2 6
In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and
AB = 30 cm, find the perimeter of the quadrilateral ARPQ.
S o l u t i o n :
It is given that P, Q and R are the mid-points of BC, CA and AB respectively.
Also, we have , and
We need to find the perimeter of quadrilateral ARPQ
In , P and R are the mid-points of CB and AB respectively.
Theorem states, the line segment joining the mid-points of any two sides of a traingle is parallel to the third side
and equal to half of it.
Therefore, we get:
Similarly, we get
We have Q and R as the mid points of AC and AB respectively.
Therefore,
And
Perimeter of 
Hence, the perimeter of quadrilateral ARPQ is .
Q u e s t i o n : 2 7
In a ?ABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.
S o l u t i o n :
 is given with AD as the median extended to point X such that .
Join BX and CX.
We get a quadrilateral ABXC, we need to prove that it’s a parallelogram.
We know that AD is the median.
By definition of median we get:
Also, it is given that
Thus, the diagonals of the quadrilateral ABCX bisect each other.
Therefore, quadrilateral ABXC is a parallelogram.
Hence proved.
Q u e s t i o n : 2 8
In a ABC, E and F are the mid-points of AC and AB respectively. The altitude AP to BC intersects FE at Q. Prove
that AQ = QP.
S o l u t i o n :
 is given with E and F as the mid points of sides AB and AC.
Also,  intersecting EF at Q.
We need to prove that 
In , E and F are the mid-points of AB and AC respectively.
Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side