Midpoint Theorem on Trapezium

PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU is drawn parallel to PQ which meets PS  at U. Prove that 2TU = PQ + RS.

Midpoint Theorem on Trapezium

Given: PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU ∥ PQ and TU meets PS at U.

To prove: 2TU = PQ + RS.

Construction: Join QS. QS and TU intersect at M.

Proof:

            Statement

            Reason

1. PQ ∥ RS and TU ∥ PQ.

1. Given.

2. RS ∥ TU.

2. From statement 1.

3. In ∆QRS,

T is the midpoint of QR and TM ∥ RS

⟹ M is the midpoint of QS.

3. By the converse of the Midpoint Theorem.

4. In ∆PSQ,

M is the midpoint of QS and MU ∥ PQ.

⟹ U is the midpoint of PS.

4. By the converse of the Midpoint Theorem.

5. In ∆QRS, the line segment TM joining the midpoints of sides QR and QS.

Therefore, TM = \(\frac{1}{2}\)RS.

5. By the Midpoint Theorem.

6. In ∆PQS, the line segment MU joins the midpoints of the sides QS and PS.

Therefore, MU = \(\frac{1}{2}\)PQ.

6. By the Midpoint Theorem.

7. TM + MU = \(\frac{1}{2}\)RS + \(\frac{1}{2}\)PQ.

7. From statements 5 and 6.

8. TU = \(\frac{1}{2}\)(RS + PQ).

8.  TM + MU = TU.


9. 2TU = RS + PQ. (Proved)

9. From statement 8.






9th Grade Math

From Midpoint 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

RSS

  1. Addition of Capacity | Add the Different Units of Capacity | Examples

    Nov 14, 24 03:03 PM

    Addition of Measurement of Capacity
    In addition of capacity we will learn how to add the different units of capacity and volume together. While adding we need to follow that the units of capacity i.e., liter and milliliter

    Read More

  2. Measuring Capacity | Standard Unit of Capacity | Litre | Millilitres

    Nov 14, 24 02:40 PM

    2 Tablespoonful of Water
    We will discuss about measuring capacity. The milkman measures milk in liters. Petrol is given in liters. Mobil oil is sold in liters. Two milk bottles contain 1 liter of milk. One milk bottle

    Read More

  3. Subtraction of Mass | Difference Between the Units of Mass | Examples

    Nov 14, 24 09:16 AM

    Subtraction of Measurement of Weight
    In subtraction of mass we will learn how to find the difference between the units of mass or weight. While subtracting we need to follow that the units of mass i.e., kilogram and gram

    Read More

  4. Worksheet on Subtraction of Mass |Word Problems on Subtraction of Mass

    Nov 13, 24 02:00 PM

    Worksheet on Subtraction of Mass
    Practice the third grade math worksheet on subtraction of mass or weight. This sheet provides different types of questions where you need to arrange the values of mass under different columns

    Read More

  5. Worksheet on Addition of Mass | Word problems on Addition of Mass

    Nov 13, 24 10:24 AM

    Practice the third grade math worksheet on addition of mass/weight. This sheet provides different types of questions where you need to arrange the values of mass under different columns

    Read More