Processing math: 100%

Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes

Here we will prove that triangles with equal areas on the same base have equal corresponding altitudes (or are between the same parallels).

Given: PQR and SQR are two triangles on the same base QR, and ar(∆PQR) = ar(∆SQC). Also, PN and SM are their corresponding altitudes.

To prove: PN = SM (or PS ∥ QR).

Construction: Join PS.

Proof:

            Statement

            Reason

1.  12 × QR × PN = 12 × QR × SM.

1. Are of a triangle = 12 × base × altitude, and ar(∆PQR) = ar(∆SQR).

2. PN = SM.

2. Cancelling 12 × QR from statement 1.

3. PN ∥ SM.

3. PN ⊥ QR and SM ⊥ QR.

4. PNMS is a rectangle.

4. PMNS is a parallelogram by statements 2 and 3, and two angles are right angles.

5. PN = SM (or PS ∥ QR). (Proved)

5. By statement 4, PNMS is a rectangle.


Corollary: Parallelograms with equal area on the same base have equal corresponding altitudes (or are between the same parallels).

Here, ar(parallelogram PQRS) = ar(parallelogram PQMN)

Therefore, ar(∆PRQ) = ar(∆PNQ)

Therefore, RN ∥ PQ. But RS ∥ PQ, NM ∥ PQ.

Therefore, RN ∥ RS and RN ∥ NM

Having common point (R or N), all the lines are coincident.

Therefore, the parallelogram have equal altitudes.





9th Grade Math

From Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes 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. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More