Chapter
6: The
Triangle and its Properties
Exercise:
6.1 (3)
Question: 1
In D is the mid-point of .
is _______________
is _________________.
Is
Solution
Given:
is altitude.
is median.
No, as is the mid-point of .
Question: 2
Draw
rough sketches for the following:
a.
In is a median.
b.
In and are altitudes of the triangle.
c.
In is an altitude in the exterior of the
triangle.
Solution
a. Here, is a median in and .
b. Here, and are the altitudes of the and .
c. is an altitude in the exterior of .
Question: 3
Verify
by drawing a diagram if the median and altitude of an isosceles triangle can be
same.
Solution
Isosceles triangle means any
two sides are same.
Take and draw the median when .
is the median and altitude of the given
triangle.
Exercise:
6.2 (2)
Question: 1
Find the value of the unknown exterior angle in the following diagrams:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
Since, Exterior
angle Sum of interior opposite angles, therefore
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Question: 2
Find
the value of the unknown interior angle in the following figures:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Exercise: 6.3 (2)
Question: 1
Find
the value of the unknown in the following diagrams:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
(i)
In
[By angle sum property of a
triangle]
(ii)
In
[By angle sum
property of a triangle]
(iii)
In
[By angle sum
property of a triangle]
(iv)
In the given isosceles triangle,
[By angle sum
property of a triangle]
(v)
In the given equilateral triangle,
[By angle sum property of a
triangle]
(vi)
In the given right angled triangle,
[By angle sum
property of a triangle]
Question: 2
Find
the values of the unknowns x and y in the following diagrams:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
(i)
[ Exterior angle property of a ]
Now, [Angle sum property of a ]
(ii)
…….(i)
[ Vertically opposite angles]
Now, [Angle sum property of a ]
[From equation (i) ]
(iii)
[Exterior angle
property of of a ]
Now, [Angle sum property of a ]
(iv)
…….(i)
[Vertically opposite angle]
Now, [Angle sum property of a ]
[From equation (i) ]
(v)
…….(i)
[Vertically opposite angle]
Now, [Angle sum property of a ]
[From equation (i)]
(vi)
…….(i)
[Vertically opposite angle]
Now, [Angle sum property of a ]
[From equation (i)]
Exercise: 6.4 (6)
Question: 1
Is it possible to have a triangle with the following
sides?
(i)
(ii)
(iii)
Solution
Since, a triangle is possible whose sum of the lengths
of any two sides would be greater than the length of third side.
(i)
No
Yes
Yes
This triangle is
not possible.
(ii)
Yes
Yes
Yes
This triangle is
possible.
(iii)
Yes
Yes
No
This triangle is not
possible
Question: 2
Take any point in the interior of a
triangle . Is:
(i)
(ii)
(iii)
Solution
Join ,
and .
(i)
Is
Yes,
form a triangle.
(ii)
Is
Yes, form a triangle.
(iii) Is
Yes, form a
triangle.
Question: 3
is a median of a triangle .
Is (Consider the sides of triangles and .)
Solution
Since, the sum of lengths of any two sides in a
triangle should be greater than the length of third side.
Therefore, In ,
…(i)
In
,
…(ii)
Adding eq. (i) and (ii),
Hence, it is true.
Question: 4
is
quadrilateral. Is
Solution
Since, the sum of lengths of any two sides in a triangle
should be greater than the length of third side.
Therefore, In ,
…(i)
In
,
…(ii)
In ,
…(iii)
In ,
…(iv)
Adding eq. (i),
(ii), (iii) and (iv), we get
Hence, it is
true.
Question: 5
is
quadrilateral. Is
Solution
Since, the sum of lengths of any two sides in a triangle
should be greater than the length of third side.
Therefore, In ,
…(i)
In
,
…(ii)
In ,
…(iii)
In ,
…(iv)
Adding equations
(i), (ii), (iii) and (iv), we get
Hence, it is
proved.
Question:
6
The lengths of two sides of a triangle are and .
Between what two measures should the length of the third side fall?
Solution
Since, the sum of lengths of any two sides in a triangle should
be greater than the length of third side.
It is given that two sides of triangle are and .
Therefore, the
third side should be less than .
And also the
third side cannot be less than the difference of the two sides.
Therefore, the
third side has to be more than .
Hence, the third side could be the length more than and less than .
Exercise: 6.5 (8)
Question: 1
is a
triangle right angled at . If cm and
cm,
find .
Solution
Given: ,
Let be .
In right angled triangle ,
[By Pythagoras
theorem]
Thus, the length of is .
Question: 2
is a triangle right angled at . If cm and cm, find .
Solution
Given: cm, cm
Let be cm.
In right angled triangle ,
[By
Pythagoras theorem]
Thus, the length of is .
Question: 3
long ladder reached a window high from the ground on placing it against a
wall at a distance .
Find the distance of the foot of the
ladder from the wall.
Solution
Let be the ladder and be the window.
Given: ,
,
In right-angled triangle ,
[By Pythagoras
theorem]
.
Thus, the distance of the foot of the ladder from the wall is
m.
Question: 4
Which
of the following can be the sides of a right triangle?
(i)
(ii)
(iii)
In
the case of right-angled triangles, identify the right angles.
Solution
Let us consider, the larger side be the hypotenuse and also
using Pythagoras theorem,
(i)
In
Since,
Therefore, the given sides
are of the right-angled triangle.
Right angle lies on the
opposite to the greater side .
i.e., at .
(ii)
In the given triangle,
Since,
Therefore, the given sides
are not of the right angled triangle.
(iii)
In
Since, .
Therefore, the given sides
are of the right-angled triangle.
Right angle lies on the
opposite to the greater side .
i.e., at .
Question: 5
tree is broken at a height of from the ground and its top touches the ground at a distance
of from the base of the tree. Find the original
height of the tree.
Solution
Let represents the tree before it got broken at
the point and let the top touches the ground at after it broke. Then is a right-angled
triangle, right angled at B.
and
Using Pythagoras theorem in ,
Hence, the total height of the tree .
Question: 6
Angles and of a are and . Write which of the following is true:
(i)
(ii)
(iii)
Solution
In
[By Angle sum
property of a ]
Thus, is a right-angled
triangle, right-angled at .
[By Pythagoras
theorem]
Hence, Option (ii) is correct.
Question:
7
Find
the perimeter of the rectangle whose length is and a diagonal is .
Solution
Given diagonal ,
length
Let breadth be cm.
Now, in right angled triangle ,
[By Pythagoras theorem]
Therefore, the breadth of the rectangle is .
Perimeter of rectangle
Hence, the perimeter of the rectangle is .
Question:
8
The diagonals of a rhombus measure and .
Find its perimeter.
Solution
Given: Diagonals and .
Since the diagonals of the rhombus bisect at right angle to each other.
Therefore, cm
And
Now, In right-angled triangle ,
[By Pythagoras theorem]
Perimeter of rhombus
Thus, the perimeter of rhombus is .