Page 175 - Maths Skills - 7

The Triangle and Its Properties 173

Let us learn more through examples.

Let’s Attempt
A
Example 1: In Fig., D is a point on the side BC of DABC. Prove that, AC + AB + BC > 2AD.
Solution: Using triangle inequality property in Ds ADC and ADB, we have
AC + CD > AD …(i)
AB + DB > AD …(ii) C D B

Adding (i) and (ii) on both the sides, we get
AC + CD + AB + DB > AD + AD
or AC + (CD + DB) + AB > 2AD [Q CD + DB = BC]
or AC + BC + AB > 2AD

Thus, AC + AB + BC > 2AD (proved).

Example 2: Can triangles be drawn with sides measuring:
(i) 1.5 cm, 3.5 cm, 6 cm (ii) 3 cm, 4 cm, 5 cm
Solution: (i) Clearly, 1.5 + 3.5 = 5 < 6

Since the sum of two of these sides is not greater than the third side, so the triangle cannot
be drawn.

(ii) Clearly, 3 + 4 = 7 > 5;
4 + 5 = 9 > 3 and 5 + 3 = 8 > 4

Since the sum of two of these sides is greater than the third side, so the triangle can be drawn.

Exercise 10.2

1. In Fig., D is a point on the side AC of DABC.

Fill in the blanks with > or < or = :
(i) BD ......... AB + AD (ii) BC + CD......... BD
1
(iii) BD ......... (AB + BC + AC)
2
2. In Fig., P and Q are the points on the side BC of DABC and AP = AQ.
Prove that AC + AB + BC > 2AP – PQ.
3. Which of the following can be the sides of a triangle?

(i) 1.8 cm, 3.5 cm, 6 cm (ii) 1 cm, 3 cm, 2 cm (iii) 1.5 cm, 2.5 cm, 5 cm
(iv) 5 cm, 7 cm, 12 cm (v) 8.5 cm, 2 cm, 5 cm (vi) 3.4 cm, 2.1 cm, 5.3 cm

4. Three points A, B, C are collinear and B is the midpoint of A and C as shown
in Fig. Can you draw a triangle ABC?

5. In Fig., is
(i) OA + OB > AB? (ii) OA + OC = AC?
(iii) OB + OC < BC?