The Sum of any Two Sides of a Triangle is Greater than the Third Side
Here we will prove that the sum of any two sides of a triangle is greater than the third side.
Given: XYZ is a triangle.
To Prove: (XY + XZ) > YZ, (YZ + XZ) > XY and (XY + YZ) > XZ
Construction: Produce YX to P such that XP = XZ. Join P and Z.
|
Statement 1. ∠XZP = ∠XPZ. 2. ∠YZP > ∠XZP. 3. Therefore, ∠YZP > ∠XPZ. 4. ∠YZP > ∠YPZ. 5. In ∆YZP, YP > YZ. 6. (YX + XP) > YZ. 7. (YX + XZ) > YZ. (Proved) |
Reason 1. XP = XZ. 2. ∠YZP = ∠YZX + ∠XZP. 3. From 1 and 2. 4. From 3. 5. Greater angle has greater side opposite to it. 6. YP = YX + XP 7. XP = XZ |
Similarly, it can be shown that (YZ + XZ) >XY and (XY + YZ) > XZ.
Corollary: In a triangle, the difference of the lengths of any two sides is less than the third side.
Proof: In a ∆XYZ, according to the above theorem (XY + XZ) > YZ and (XY + YZ) > XZ.
Therefore, XY > (YZ - XZ) and XY > (XZ - YZ).
Therefore, XY > difference of XZ and YZ.
Note: Three given lengths can be sides of a triangle if the sum of two smaller lengths greater than the greatest length.
For example: 2 cm, 5 cm and 4 cm can be the lengths of three sides of a triangle (since, 2 + 4 = 6 > 5). But 2 cm, 6.5 cm and 4 cm cannot be the lengths of three sides of a triangle (since, 2 + 4 ≯ 6.5).
From The Sum of any Two Sides of a Triangle is Greater than the Third Side to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.