Properties of Triangles (Exercise 15.4) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

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1. In each of the following, there are three positive numbers. State if these numbers 
could possibly be the lengths of the sides of a triangle: 
(i) 5, 7, 9 
(ii) 2, 10, 15 
(iii) 3, 4, 5 
(iv) 2, 5, 7 
(v) 5, 8, 20 
 
Solution: 
(i) Given 5, 7, 9 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of a triangle is always greater than the third side.  
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5 
 
(ii) Given 2, 10, 15 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case. 
Here, 2 + 10 < 15 
 
(iii) Given 3, 4, 5 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of triangle is always greater than the third side.  
Here, 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3 
 
(iv) Given 2, 5, 7 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
Here, 2 + 5 = 7 
 
(v) Given 5, 8, 20 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
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1. In each of the following, there are three positive numbers. State if these numbers 
could possibly be the lengths of the sides of a triangle: 
(i) 5, 7, 9 
(ii) 2, 10, 15 
(iii) 3, 4, 5 
(iv) 2, 5, 7 
(v) 5, 8, 20 
 
Solution: 
(i) Given 5, 7, 9 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of a triangle is always greater than the third side.  
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5 
 
(ii) Given 2, 10, 15 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case. 
Here, 2 + 10 < 15 
 
(iii) Given 3, 4, 5 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of triangle is always greater than the third side.  
Here, 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3 
 
(iv) Given 2, 5, 7 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
Here, 2 + 5 = 7 
 
(v) Given 5, 8, 20 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
 
 
 
 
 
 
 
Here, 5 + 8 < 20 
 
2. In Fig. 46, P is the point on the side BC. Complete each of the following statements 
using symbol ‘=’,’ > ‘or ‘< ‘so as to make it true: 
(i) AP… AB+ BP 
(ii) AP… AC + PC 
(iii) AP…. ½ (AB + AC + BC) 
 
Solution: 
(i) In ?APB, AP < AB + BP because the sum of any two sides of a triangle is greater than 
the third side. 
 
(ii) In ?APC, AP < AC + PC because the sum of any two sides of a triangle is greater than 
the third side. 
 
(iii) AP < ½ (AB + AC + BC)  
In ?ABP and ?ACP, we can write as 
AP < AB + BP… (i) (Because the sum of any two sides of a triangle is greater than the 
third side) 
AP < AC + PC … (ii) (Because the sum of any two sides of a triangle is greater than the 
third side) 
On adding (i) and (ii), we have: 
AP + AP < AB + BP + AC + PC 
2AP < AB + AC + BC (BC = BP + PC) 
AP < ½ (AB + AC + BC) 
  
3. P is a point in the interior of ?ABC as shown in Fig. 47. State which of the following 
statements are true (T) or false (F): 
(i) AP + PB < AB 
(ii) AP + PC > AC 
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1. In each of the following, there are three positive numbers. State if these numbers 
could possibly be the lengths of the sides of a triangle: 
(i) 5, 7, 9 
(ii) 2, 10, 15 
(iii) 3, 4, 5 
(iv) 2, 5, 7 
(v) 5, 8, 20 
 
Solution: 
(i) Given 5, 7, 9 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of a triangle is always greater than the third side.  
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5 
 
(ii) Given 2, 10, 15 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case. 
Here, 2 + 10 < 15 
 
(iii) Given 3, 4, 5 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of triangle is always greater than the third side.  
Here, 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3 
 
(iv) Given 2, 5, 7 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
Here, 2 + 5 = 7 
 
(v) Given 5, 8, 20 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
 
 
 
 
 
 
 
Here, 5 + 8 < 20 
 
2. In Fig. 46, P is the point on the side BC. Complete each of the following statements 
using symbol ‘=’,’ > ‘or ‘< ‘so as to make it true: 
(i) AP… AB+ BP 
(ii) AP… AC + PC 
(iii) AP…. ½ (AB + AC + BC) 
 
Solution: 
(i) In ?APB, AP < AB + BP because the sum of any two sides of a triangle is greater than 
the third side. 
 
(ii) In ?APC, AP < AC + PC because the sum of any two sides of a triangle is greater than 
the third side. 
 
(iii) AP < ½ (AB + AC + BC)  
In ?ABP and ?ACP, we can write as 
AP < AB + BP… (i) (Because the sum of any two sides of a triangle is greater than the 
third side) 
AP < AC + PC … (ii) (Because the sum of any two sides of a triangle is greater than the 
third side) 
On adding (i) and (ii), we have: 
AP + AP < AB + BP + AC + PC 
2AP < AB + AC + BC (BC = BP + PC) 
AP < ½ (AB + AC + BC) 
  
3. P is a point in the interior of ?ABC as shown in Fig. 47. State which of the following 
statements are true (T) or false (F): 
(i) AP + PB < AB 
(ii) AP + PC > AC 
 
 
 
 
 
 
 
(iii) BP + PC = BC 
 
Solution: 
(i) False 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
not true for the given triangle. 
 
(ii) True 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
true for the given triangle. 
 
(iii) False 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
not true for the given triangle. 
 
4. O is a point in the exterior of ?ABC. What symbol ‘>’,’<’ or ‘=’ will you see to 
complete the statement OA+OB….AB? Write two other similar statements and show 
that OA + OB + OC > ½ (AB + BC +CA) 
 
Solution: 
We know that the sum of any two sides of a triangle is always greater than the third 
side, in ?OAB, we have, 
OA + OB > AB ….. (i) 
In ?OBC we have 
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1. In each of the following, there are three positive numbers. State if these numbers 
could possibly be the lengths of the sides of a triangle: 
(i) 5, 7, 9 
(ii) 2, 10, 15 
(iii) 3, 4, 5 
(iv) 2, 5, 7 
(v) 5, 8, 20 
 
Solution: 
(i) Given 5, 7, 9 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of a triangle is always greater than the third side.  
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5 
 
(ii) Given 2, 10, 15 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case. 
Here, 2 + 10 < 15 
 
(iii) Given 3, 4, 5 
Yes, these numbers can be the lengths of the sides of a triangle because the sum of any 
two sides of triangle is always greater than the third side.  
Here, 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3 
 
(iv) Given 2, 5, 7 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
Here, 2 + 5 = 7 
 
(v) Given 5, 8, 20 
No, these numbers cannot be the lengths of the sides of a triangle because the sum of 
any two sides of a triangle is always greater than the third side, which is not true in this 
case.  
 
 
 
 
 
 
 
Here, 5 + 8 < 20 
 
2. In Fig. 46, P is the point on the side BC. Complete each of the following statements 
using symbol ‘=’,’ > ‘or ‘< ‘so as to make it true: 
(i) AP… AB+ BP 
(ii) AP… AC + PC 
(iii) AP…. ½ (AB + AC + BC) 
 
Solution: 
(i) In ?APB, AP < AB + BP because the sum of any two sides of a triangle is greater than 
the third side. 
 
(ii) In ?APC, AP < AC + PC because the sum of any two sides of a triangle is greater than 
the third side. 
 
(iii) AP < ½ (AB + AC + BC)  
In ?ABP and ?ACP, we can write as 
AP < AB + BP… (i) (Because the sum of any two sides of a triangle is greater than the 
third side) 
AP < AC + PC … (ii) (Because the sum of any two sides of a triangle is greater than the 
third side) 
On adding (i) and (ii), we have: 
AP + AP < AB + BP + AC + PC 
2AP < AB + AC + BC (BC = BP + PC) 
AP < ½ (AB + AC + BC) 
  
3. P is a point in the interior of ?ABC as shown in Fig. 47. State which of the following 
statements are true (T) or false (F): 
(i) AP + PB < AB 
(ii) AP + PC > AC 
 
 
 
 
 
 
 
(iii) BP + PC = BC 
 
Solution: 
(i) False 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
not true for the given triangle. 
 
(ii) True 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
true for the given triangle. 
 
(iii) False 
Explanation: 
We know that the sum of any two sides of a triangle is greater than the third side, it is 
not true for the given triangle. 
 
4. O is a point in the exterior of ?ABC. What symbol ‘>’,’<’ or ‘=’ will you see to 
complete the statement OA+OB….AB? Write two other similar statements and show 
that OA + OB + OC > ½ (AB + BC +CA) 
 
Solution: 
We know that the sum of any two sides of a triangle is always greater than the third 
side, in ?OAB, we have, 
OA + OB > AB ….. (i) 
In ?OBC we have 
 
 
 
 
 
 
 
OB + OC > BC …… (ii) 
In ?OCA we have 
OA + OC > CA ….. (iii) 
On adding equations (i), (ii) and (iii) we get: 
OA + OB + OB + OC + OA + OC > AB + BC + CA 
2(OA + OB + OC) > AB + BC + CA 
OA + OB + OC > (AB + BC + CA)/2 
Or  
OA + OB + OC > ½ (AB + BC +CA) 
Hence the proof. 
 
5. In ?ABC, ?A = 100
o
, ?B = 30
o
, ?C = 50
o
. Name the smallest and the largest sides of 
the triangle. 
 
Solution: 
We know that the smallest side is always opposite to the smallest angle, which in this 
case is 30°, it is AC.  
Also, because the largest side is always opposite to the largest angle, which in this case 
is 100°, it is BC.