The document Ex 7.4 NCERT Solutions- Triangles Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.

All you need of Class 9 at this link: Class 9

**Question 1. Show that in a right angled triangle, the hypotenuse is the longest side Solution:** Let us consider Î”ABC such that âˆ B = 90Âº

âˆ´ âˆ A + âˆ B + âˆ C = 180Âº

âˆ´ [âˆ A + âˆ C] + âˆ B = 180Âº

â‡’ âˆ A + âˆ C = 90Âº

â‡’ âˆ A + âˆ C= âˆ B

âˆ´ âˆ B > âˆ A and âˆ B > âˆ C

â‡’ Side opposite to âˆ B is longer than the side opposite to âˆ A.

i.e. AC > BC ...(1)

Similarly, AC > AB ...(2)

From (1) and (2), we get AC is the longest side.

But AC is the hypotenuse of the triangle.

Thus, hypotenuse is the longest side.

**Question 2. In the adjoining figure, sides AB and AC of Î”ABC are extended to points P and Q respectively. Also, âˆ PBC < âˆ QCB. Show that AC > AB. Solution:** âˆ ABC + âˆ PBC = 180Âº [Linear pair]

and âˆ ACB + âˆ QCB = 180Âº [Linear pair]

âˆ´ âˆ ABC + âˆ PBC = âˆ ACB + âˆ QCB

But âˆ PBC < âˆ QCB [Given]

âˆ´ âˆ ABC > âˆ ACB

â‡’ [The side opposite to âˆ ABC] > [The side opposite to âˆ ACB]

â‡’ AC > AB

**Question 3. In the figure, âˆ B < âˆ A and âˆ C < âˆ D. Show that AD > BC. Solution:** âˆµ âˆ B< âˆ A [Given]

â‡’ âˆ A> âˆ B

âˆ´ OB > OA [Side opposite to greater angle is longer] ...(1)

Similarly, OC > OD ...(2)

From (1) and (2), we have

[OB + OC] > [OA + OD]

â‡’ BC > AD or AD < BC

**Question 4. AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see figure). Show that âˆ A > âˆ C and âˆ B > âˆ D. Solution:** Let us join AC.

Now, in Î”ABC, AB < BC [âˆµ AB is the smallest side of quadrilateral ABCD]

â‡’ BC > AB

âˆ´ [Angle opposite to BC] < [Angle opposite to AB]

â‡’ âˆ BAC > âˆ BCA ...(1)

Again, in Î”ACD,

CD > AD [âˆµ CD is the longest side of the quadrilateral ABCD]

âˆ´ [Angle opposite to CD] > [Angle opposite to AD]

â‡’ âˆ CAD > âˆ ACD ...(2)

Adding (1) and (2), we get

[âˆ BAC + CAD] > [âˆ BCA + âˆ ACD]

â‡’ âˆ A> âˆ C

Similarly, by joining BD, we have âˆ B> âˆ D

**Question 5. In the figure, PR > PQ and PS bisects âˆ QPR. Prove that âˆ PSR > âˆ PSQ. Solution: **In Î”PQR, PS bisects âˆ QPR [Given]

âˆ´ âˆ QPS = âˆ RPS

âˆµ PR > PQ [Given]

âˆ´ [Angle opposite to PR] > [Angle opposite to PQ]

â‡’ âˆ PQS > âˆ PRS

â‡’ [âˆ PQS + âˆ QPS] > [âˆ PRS + âˆ RPS] ...(1)

[âˆµ âˆ QPS = âˆ RPS]

âˆµ Exterior âˆ PSR = [âˆ PQS + âˆ QPS]

[âˆµ An exterior angle is equal to the sum of interior opposite angles] and

Exterior âˆ PSQ = [âˆ PRS + âˆ RPS]

Now, from (1), we have âˆ PSR > âˆ PSQ**Question 6. Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest. Solution:** Let us consider the Î”PMN such that âˆ M = 90Âº

Since, âˆ M + âˆ N + âˆ P = 180Âº

[Sum of angles of a triangle] âˆµ

âˆ M = 90Âº [âˆµ PM âŠ¥ â„“]

â‡’ âˆ N< âˆ M â‡’ PM < PN ...(1)

Similarly, PM < PN_{1} ...(2)

PM < PN_{2 } ...(3)

From (1), (2) and (3), we have PM is the smallest line segment drawn from P on the line l.

Thus, the perpendicular segment is the shortest line segment drawn on a line from a point not on it.

192 videos|230 docs|82 tests

### Some Properties of a Triangle Theorems(Hindi)

- Video | 05:47 min
### Ex 7.5 NCERT Solutions- Triangles

- Doc | 2 pages
### Some Properties of a Triangle Theorems

- Video | 07:58 min
### Meaning of Corresponding Parts of Congruent Triangles

- Video | 03:48 min
### Congruence of Triangles(Hindi)

- Video | 06:17 min
### Examples: Properties of Triangles

- Video | 13:21 min

- Test:Triangles- 2
- Test | 25 ques | 25 min
- What are Congruent Figures?(Hindi)
- Video | 09:53 min