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.

Midsegment Theorem on Trapezium

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.






9th Grade Math

From Midsegment Theorem on Trapezium 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

  1. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Oct 22, 24 03:26 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  2. Word Problems on Multiplication |Multiplication Word Problem Worksheet

    Oct 22, 24 01:23 AM

    Multiplication Word Problem
    Word problems on multiplication for fourth grade students are solved here step by step. Problem Sums Involving Multiplication: 1. 24 folders each has 56 sheets of paper inside them. How many sheets of…

    Read More

  3. Worksheet on Word Problems on Multiplication | Multiplication Problems

    Oct 22, 24 12:31 AM

    In worksheet on word problems on multiplication, all grade students can practice the questions on word problems involving multiplication. This exercise sheet on word problems on multiplication

    Read More

  4. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Oct 21, 24 03:38 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More

  5. Multiplication Table of 4 |Read and Write the Table of 4|4 Times Table

    Oct 21, 24 02:26 AM

    Multiplication Table of Four
    Repeated addition by 4’s means the multiplication table of 4. (i) When 5 candle-stands having four candles each. By repeated addition we can show 4 + 4 + 4 + 4 + 4 = 20 Then, four 5 times

    Read More