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Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q. (1) In fig (10).9(2) It is given that AB = CD and AD = BC. Prove that ΔADC≅ΔCBA.

Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Solution:

Given that in the figure AB = CD  and AD = BC.

We have to prove ΔADC≅ΔCBA

Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Now,

Consider Δ ADC and Δ CBA.

We have

AB = CD                     [Given]

BC = AD                     [Given]

And AC = AC             [Common side]

So, by SSS congruence criterion, we have

ΔADC≅ΔCBA

Hence proved

 

Q. (2) In a Δ PQR. IF PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.

Sol: Given that in Δ PQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively

We have to prove LN = MN.

Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

Join L and M, M and N, N and L

We have PL = LQ, QM = MR and RN = NP

[Since, L, M and N are mid-points of Pp. QR and RP respectively]

And also PQ = QR

  • PL = LQ = QM = MR = Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics ……(i) Using mid-point theorem,

We have

MN ∥ PQ and MN = Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics

  • MN = PL = LQ ……(ii)

Similarly, we have

LN ∥  QR and LN = (1/2)QR

  • LN = QM = MR ……(iii)

From equation (i), (ii) and (iii), we have

PL = LQ = QM = MR = MN = LN

LN = MN

The document Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions | RD Sharma Solutions for Class 9 Mathematics is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
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FAQs on Ex-10.4, Congruent Triangles, Class 9, Maths RD Sharma Solutions - RD Sharma Solutions for Class 9 Mathematics

1. What are congruent triangles?
Ans. Congruent triangles are triangles that have the same shape and size. In other words, all corresponding sides and angles of the triangles are equal.
2. How do we prove two triangles congruent?
Ans. Two triangles can be proven congruent using different methods such as Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) congruence criteria. These criteria compare the corresponding sides and angles of the triangles to establish their congruence.
3. What is the importance of congruent triangles in mathematics?
Ans. Congruent triangles are important in mathematics as they help in solving various geometrical problems. They allow us to establish equalities between different parts of the triangles, which can be used to find unknown angles or side lengths. Moreover, congruent triangles serve as a foundation for more advanced geometric concepts.
4. Can two triangles with all sides equal be congruent?
Ans. Yes, two triangles with all sides equal can be congruent. This is known as the Side-Side-Side (SSS) congruence criterion. If all three sides of one triangle are equal to the corresponding sides of another triangle, then the triangles are congruent.
5. How are congruent triangles used in real-life situations?
Ans. Congruent triangles have various applications in real-life situations. They are used in architecture and construction to ensure that buildings and structures are built accurately. Congruent triangles are also used in map-making and navigation, where they help in determining distances and angles between different locations. Additionally, congruent triangles are used in engineering and design to create symmetrical and balanced objects.
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