# Midsegment Theorem on Trapezium

Here we will prove that the line segment joining the midpoints of the nonparallel sides of a trapezium is half the sum of the lengths of the parallel sides and is also parallel to them.

Solution:

Given: PQRS is a trapezium in which PQ ∥ RS. U and V are the midpoints of QR and PS respectively.

To prove: (i) UV ∥ RS.

(ii) UV = $$\frac{1}{2}$$(PQ + RS).

Construction: Join QV and produce it to meet RS produced at T.

Proof:

 Statement Reason 1. In ∆PQV and ∆STV,(i) PV = VS.(ii) ∠PVQ = ∠TVS.(iii) ∠QPV = ∠VST. 1.(i) Given.(ii) Vertically opposite angles.(iii) Alternate angles. 2. Therefore, ∆PQV ≅ ∆STV. 2. By ASA criterion of congruency. 3. Therefore, PQ = ST. 3. CPCTC. 4. QV = VT. 4. CPCTC. 5. In ∆QRT,(i) U is the midpoint of QR.(ii) V is the midpoint of QT. 5.(i) Given.(ii) From statement 4. 6. Therefore, UV ∥ RT and UV = $$\frac{1}{2}$$RT. 6. By the Midpoint Theorem. 7. Therefore, UV = $$\frac{1}{2}$$(RS+ ST). 7. From statement 6. 8. UV = $$\frac{1}{2}$$(RS+ PQ). 8. Using statement 3 in statement 7. 9. Therefore, UV ∥ RS and UV = $$\frac{1}{2}$$(PQ+ RS). (Proved) 9. From statement 6 and 8.

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.

## Recent Articles

1. ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

Jul 12, 24 03:08 PM

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

2. ### Worksheet on Fractions | Questions on Fractions | Representation | Ans

Jul 12, 24 02:11 PM

In worksheet on fractions, all grade students can practice the questions on fractions on a whole number and also on representation of a fraction. This exercise sheet on fractions can be practiced

3. ### Fraction in Lowest Terms |Reducing Fractions|Fraction in Simplest Form

Jul 12, 24 03:21 AM

There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1…

4. ### Conversion of Improper Fractions into Mixed Fractions |Solved Examples

Jul 12, 24 12:59 AM

To convert an improper fraction into a mixed number, divide the numerator of the given improper fraction by its denominator. The quotient will represent the whole number and the remainder so obtained…