Q1 )
In quadrilateral ACBD, AC = AD and AB bisects
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AC = AD ....(Given)
As, AB bisects
Therefore,
Also, AB is a common side,
So, we can say,
Also, BC = BD ...(By CPCT)
Hence, proved.
Q2 )
ABCD is a quadrilateral in which AD = BC and
(i)
(ii) BD = AC
(iii)
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AD = BC ...(Given)
Also,
And AB is a common side.
Therefore,
Hence, BD = AC ...(By CPCT)
and
Hence, proved.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
Now,
Also,
and BD = BC ...(shown in the figure)
Therefore,
Hence, OA = OB ...(By CPCT)
Thus, we can say that, O is the mid-point of AB.
So, CD bisects AB.
Hence, proved
Q4 )
l and m are two parallel lines intersected by another pair of parallel lines p and q (see figure). Show that
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
From figure, we have,
From Equations, (i), (ii) and (iii), we have
In
and AC is a common side.
Therefore, we get,
Hence, proved.
Q5 )
Line l is the bisector of a
(i)
(ii) BP = BQ or B is equidistant from the arms of
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB is the common side.
Also, BP = BQ ...(By CPCT)
Thus, we can say that, B is equidistant from the arms of
Hence, proved.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB = AD ...(Given)
On adding,
and also, AC = AE ...(Given)
Thus, BC = DE ...(By CPCT)
Hence, proved.
Q7 )
AB is a line segment and P is its mid-point. D and E are points on the same side of
AB such that
(i)
(ii) AD = BE
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
We have,
AP = BP ...(i)(Since, P is the mid-point of AB)
On adding,
we have,
Now, In
and AP = BP ...(From Eq. (i))
Thus, AD = BE ...(By CPCT)
Hence, proved.
Q8 )
In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is
joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see figure).
Show that
(i)
(ii)
(iii)
(iv)CM = (1/2) AB
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
Given :
To prove :
(i)
(ii)
(iii)
(iv)CM = (
Construction : Produce CM to D, such that CM = MD. Join DB.
Proof : In
AM = BM ...(M is the mid-point of AB)
CM = DM ...(Given)
and
Hence, part(i) is proved.
Also, AC = DB ...(By CPCT)
But,
Hence, part(ii) is proved, too.
Now, considering
AC = DB ...(from part(i))
Side BC is common.
and
Therefore,
Hence, now, part(iii) is proved, too.
Now, DC = AB ...(By CPCT)
Multipling both sides by
(
Now, as we know, CM = (
Therefore, CM = (
Hence, part (iv) is proved.
Q1 )
In an isosceles triangle ABC, with AB = AC, the bisectors of
(i) OB = OC
(ii) AO bisects
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB = AC ...(Given)
Also,
Multipling both sides by
(
Also, it is given that, OB and OC are bisectors of
Therefore,
Therefore, OB = OC
...(
Hence, part (i) is proved.
In
AB = AC ...(Given)
OB = OC ...(proved earlier)
Therefore,
Thus,
Hence, AO bisects
Q2 )
In the
Show that
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
DB = DC ...(given)
and AD is the Common side.
Therefore,
Therefore, AB = AC ...(By CPCT)
Hence,
Hence, proved.
Q3 )
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides
AC and AB respectively (see figure).
Show that these altitudes are equal.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
Also,
and AB = AC ...(Given)
Therefore,
Therefore, AB = AC ...(By CPCT)
Hence, BE = CF ...(By CPCT)
Hence, proved.
Q4 )
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see
figure). Show that
(i)
(ii) AB = AC i.e., ABC is an isosceles triangle.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
and BE = CF ...(Given)
Thus, AB = AC ...(By CPCT)
Hence, proved.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB = AC ...(
Similarly, in
BD = CD ...(
Now, On adding, Equations (i) and (ii), we get,
Hence, proved.
Q6 )
Show that
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB = AC ...(i)(Given)
Also, AB = AD ...(iii)(Given)
From (i) and (iii), AC = AD
Now, in
AD = AC ...(proved earlier)
Also,
On adding Equations (ii) and (iv), we get,
Adding
Therefore,
Hence, proved.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
In
AB = AC ...(Given)
Now, we know that,
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
Let
AB = BC = CA (by property)
Now, we have,
AB = AC
Similarly,
CB = CA
From Equations (i),(ii) and (iii), we have,
Hence, proved.
Q1 )
(i)
(ii)
(iii) AP bisects
(iv) AP is the perpendicular bisector of BC.
NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Answer :
Given:
Proof:
In
AB = AC ...(Given)
BD = CD ...(Given)
and AD is Common side.
Therefore,
Hence, part (i) is proved.
In
AB = AC ...(Given)
Also,
and AP is the common side.
Therefore,