Sum Of Any Two Sides Is Greater Than Twice The Median

Here we will prove that in a triangle the sum of any two sides is greater than twice the median which bisects the remaining side.

Solution:

Given: In ∆XYZ, XP is the median that bisects YZ at P.

SAS Criterion of Congruency

To prove: (XY + XZ) > 2XP.

Construction: Produce XP to Q such that XP = PQ. Join Z and Q.

Sum Of Any Two Sides Is Greater Than Twice The Median

Proof:

            Statement

          Reason

1. In ∆XYP and ∆ZPQ,

(i) YP = PZ.

(ii) XP = PQ

(iii) ∠XPY = ∠ZPQ

1.

(i) XP bisects YZ.

(ii) By construction.

(iii) Vertically opposite angles.

 2. XYP ≅ ∆ZPQ

2. By SAS criterion of congruency.

3. XY = ZQ.

3. CPCTC.

4. In ∆XZQ, (XZ + ZQ) > XQ.

4. Sum of the two sides of a triangle is greater than the third side.

5. (XZ + XY) > (XP + PQ).

5. XY = ZQ, from statement 3.

6. (XY + XZ) > 2 XP. (Proved)

6. XP = PQ.






9th Grade Math

From Sum Of Any Two Sides Is Greater Than Twice The Median 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.




Share this page: What’s this?

Recent Articles

RSS

  1. 2nd Grade Geometry Worksheet | Plane and Solid Shapes | Point | Line

    Dec 10, 24 02:35 PM

    Curved Line and Straight Line
    2nd grade geometry worksheet

    Read More

  2. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Dec 09, 24 10:39 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More

  3. Points and Line Segment | Two Points in a Curved Surface | Curve Line

    Dec 09, 24 01:08 AM

    Curved Lines and Straight Line
    We will discuss here about points and line segment. We know when two lines meet we get a point. When two points on a plane surface are joined, a straight line segment is obtained.

    Read More

  4. Solid Shapes | Basic Geometric Shapes | Common Solid Figures | Plane

    Dec 08, 24 11:19 PM

    Solid Shapes
    We will discuss about basic solid shapes. We see a variety of solid objects in our surroundings. Solid objects have one or more shapes like the following. Match the objects with similar shape.

    Read More

  5. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 07, 24 03:38 PM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More