Properties of Triangles

Exercise 6.4

Question 1: Is it possible to have a triangle with the following sides?

  1. 2 cm , 3 cm, 5 cm
  2. 3 cm, 6 cm, 7 cm
  3. 6 cm, 3 cm, 2 cm

Answer: Making a triangle is possible only with sides given in option ‘b’. In other options, sum of two sides is either equal to or less than the third side.


Question 2: Take any point O in the interior of a triangle PQR. Is

triangles
  1. OP + OQ > PQ?
  2. OQ + OR > QR?
  3. OR + OP > RP?

Answer: The answer is yes in each case, because sum of any two sides of a triangle is always greater than the third side.

Question 3: AM is a median of a triangle ABC. Is AB + BC + CA > 2AM? (Consider the sides of ΔABM and ΔAMC)

triangles

Answer: In ΔABM, AB + BM > AM

Similarly, in ΔAMC, AC + CM > AM

Adding above equations, we get,

AB + BM + CM + AC > 2AM

Or, AB + BC + CA > 2AM


Question 4: ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

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Answer: In ΔABC; AB + BC > AC

In ΔDAC, DA + CD > AC

In ΔDAB, DA + AB > DB

In ΔDCB, CD + CB > DB

Adding above equations, we get,

2AB + 2BC + 2 CD + 2AD > 2AC + 2BD

Or, 2(AB + BC + CD + AD) > 2(AC + BD)

Or, AB + BC + CD + AD > AC + BD


Question 5: ABCD is a quadrilateral. Is AB + BC + CD + DA < 2(AC + BD)?

Answer: Let us assume a point O at the point of intersection of diagonals AC and BD.

In ΔOAB, OA + OB > AB

In Δ OBC, OB + OC > BC

In ΔODC, OD + OC > CD

In ΔOAD, OD + OA > AD

Adding above equations, we get;

AB + BC + CD + DA < OA + OB + OB + OC + OC + OD + OD + OA

Or, AB + BC + CD + DA < OA + OA + OC + OC + OD + OD + OB + OB

Or, AB + BC + CD + DA < 2(OA + OC + OD + OB)

Or, AB + BC + CD + DA < 2(AC + BD)

Question 6: The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?

Answer: Sum of given two sides = 12 cm + 15 cm = 27 cm

Hence, the third side should always be less than 27 cm.

The difference between given sides = 15 cm – 12 cm = 3 cm

If the third side will be = 3 cm then 12 + 3 = 15 cm shall be equal to one of the given sides.

Hence, the third side should be more than 3 cm

So, range of measure of third side = 4 cm to 26 cm.



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