# NCERT Class 10 Chapter 7 Triangles Official CBSE Board Sample Problems Long Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Question

In , if . then prove that . Also, if , and , then find DE.

### Solution

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

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

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

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

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

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)

(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 .

### 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)

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

### Solution

Given: and

PM is the median of

.

To Prove:-

Proof:

Similarly, PM is the median

Now,

.

(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,

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

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.

## 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)

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