NCERT Class 10 Chapter 7 Triangles Official CBSE Board Sample Problems Long Answer
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Question
In , if . then prove that . Also, if , and , then find DE.
Solution

Triangles
Given: . i.e.
To prove:
Proof: In and
[Given]
[Common]
So, [By AA similarity]
Hence, .
Question
In, from A and B altitudes AD and BE are drawn. Prove that. Is and
Solution

Triangles
In and ,
(Each )
(Common)
So, (By AA similarity)
is not similar to.
and is not similar to .
Question
State and prove Converse of Pythagoras’ Theorem.
Solution

Triangles
Statement: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is right angle.
Given: In
To prove:
Construction: Construct a , such that and and
Proof: In , (Given)
(By Pythagoras)
(Using and by construction) ..(i)
but (Given) ...(ii)
Equating (1) and (ii)
Now, in and
(By construction)
(by construction)
(Proved)
(By SSS)
Hence
This shows that, MBC is an right-angled triangle.
Question
Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then other two sides are divided in the same ratio.
Solution

Triangles
Given: A triangle ABC in which
and DE interacts AB in D and AC in E.
To prove-
Construction: Join BE, CD and draw and
Proof: is perpendicular to AB.
is height of triangles ADE and DBE
Now, .. (i)
Similarly. So is height of and .
..(ii)
Ut, and arc on same basc DE and between same parallels DE and BC. ar(DBE) •
Multiplying both side by
Hus, [from (i) and (ii)]
Question
In and Y is middle point of BC. Then prove that,
(i)
(ii)
Solution
Given: A in which and Y is mid-point of BC.
To prove: (i)
(ii)
Proof: (i) In .
[BY Pythagoras Theorem]
(Y is midpoint of BC)
[In]
Hence proved.
(ii) In ,
[By Pythagoras Theorem]
: In Y is mid-point of BCI
(... In ]
Hence proved
Question
Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
Solution

Triangles
Given: Two triangles ABC and DEF such that .
To prove:
Construction: Draw and
Proof: In and,
[BY Construction]
:. By AA criterion of similarity,
So,
...(II) (Ratio of corresponding sides of similar triangles arc equal]
From (I) and (ii), we get
...(iii)
Now,
(From (iii)]
And
Hence,
Hence proved
Question
O is any point inside a rectangle ABCD. Prove that:
.
Solution

Rectangle
Through O, draw so that P lies on
AB and Q lies on DC.
Now,
Therefore, and ( and ( )
So, and
Therefore, BPQC and APQD are both rectangles.
Now, from ,
…(1)
Similarly, Irvin
…(2)
From , we have
… (3)
and from . we have
…. (4)
Adding (1) and (2),
(As BP=CQ and DQ=AP)
[From (3) and (4)]
Question
In the given figure, line segment XY is parallel to side AC of triangle ABC and it divides the Triangle into two parts of equal areas Find the ratio .

Triangle
Solution
In figure, the line segment XY is parallel to side AC of ABC and it divides the triangle into two parts of equal areas. Find the ratio
Given:
XY is parallel to AC i.e.
To find:
Proof:
Ln ,
(Common)
(Since XV I AC, corresponding angles are equal)
(AA similarity)

Triangle
Now,
We know that in similar triangles,
Ratio of area of triangle is equal to ratio of square of corresponding sides
(As Area of =)
Question
AD and PM are medians of triangles ABC and PQR respectively, where . Prove that

Triangle
Solution
Given: and
AD is the median of
PM is the median of
.
To Prove:-
Proof:
Since AD is the median
Similarly, PM is the median
Now,

Triangle
.
(Corresponding sides of similar triangle ore proportional)
So,
(Since AD & PM are medians)
… (1)
PQ QM
Also, since.
(Corresponding angles of similar triangles ate equal)(2)
Now,

Triangle
In
(From (2)
(From (1)
Hence by SAS similarly
Since corresponding sides of similar triangles are proportional
Hence proved
Question
The perpendicular drawn from A on side BC of a intersects BC at D such that Prove that
Solution

Triangle
Given: with
Also
To prove:
Proof:
Let
(As DB=3CD given) =3x
and
Equating the two values of we get:
Question
D, E and F are respectively the mid points of the sides AB. BC and CA of triangle ABC respectively. Find the ratio of areas of triangle DEF and triangle ABC.
Solution
Question
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
Solution
Let ABCD be a square and each side be a units
Let
and
be the equilateral triangles described on the side BC and diagonal AC respectively
(Pythagoras Theorem)
(All equilateral triangles are similar by AAA)