Triangles on the Same Base and between the Same Parallels are Equal in Area

Here we will prove that triangles on the same base and between the same parallels are equal in area.

Given: PQR and SQR are two triangles on the same base QR and are between the same parallel lines QR and MN, i.e., P and S are on MN.

To prove: ar(∆PQR) = ar(∆SQR).

Construction: Draw QM RP cutting MN at M.

Proof:

            Statement

            Reason

1. QRPM is a parallelogram.

1. MP ∥ QR and QM ∥ RP by construction.

2. ar(∆PQR) = 12 × ar(parallelogram QRPM).

ar(∆SPQ) = 12 × ar(parallelogram QRPM).

2. Area of a triangle = 12 × area of a parallelogram, on the same base, and between the same parallels.

3. ar(∆PQR) = ar(∆SQR). (Proved)

3. From statements in 2.

Corollaries:

(i) Triangles with equal bases and between the same parallels are equal in area.

(ii) If two triangles have equal bases, ratio of their areas = ratio of their altitudes.

(iii) If two triangles have equal altitudes, ratio of their areas = ratio of their bases.

(iv) A median of a triangle divides the triangle in two triangles of equal area.








9th Grade Math

From Triangles on the Same Base and between the Same Parallels are Equal in Area 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. 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