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

In the previous articles, we have learned about the theorem Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels. We know that the area of the triangle is half of the product of base and height. Here we will discuss proving the triangles on the Same Base and between the Same Parallels are Equal in Area with step by step explanation here.

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

Theorem:
Prove that Triangles on the Same Base and between the Same Parallel are Equal in Area.
To prove that:
Triangles on the same and between the same parallel are equal in area.
Proof:
△PQR & △PQS are drawn on the same base AB as base lying between the same two parallel lines L1 and L2
Here we will be using the following property of a triangle and parallelogram drawn on the same base between two parallel lines.
Since △PQR and parallelogram PQAB are on the same base and between the same parallel L1 and L2.
Therefore, ar(△PQR)= ½ × area of parallelogram PQAB.
Similarly,
ar(△PQS)= ½ × area of parallelogram PQAB.
Therefore, the area of △PQR=area of △PQS.
Hence proved

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

Example 1.
Show that a median of a triangle divides it into two triangles of equal areas.
Solution:
Let PQR be a triangle and let PS be one of its medians.
Show that
ar (PQS) = ar (PRS).
The formula for area involves an altitude let us draw PO perpendicular to QR
Now
ar(PQS) = ½ × base × altitude ( ∆PQS)
= ½ × QS × PO
= ½ × base × altitude ( ∆PRS) = ar(PRS)

Example 2.
The ratio of the areas of two triangles of the same base is equal to the ratio of their heights.
Solution:
Consider the figure
Here
Area(ΔPQR)= ½ × QM × PR
And, Area(ΔPSR)= ½ ×SN×PR
Area(∆PQR) / Area(ΔPSR) = ½ × QM × PR / ½ × SN × AC
​= QM / SN
​Hence proved

Example 3.
In the right angle triangle ∠R=90°.PT and QS are two medians of a triangle PQR meeting at U. The ratio of the area of △PQU and the quadrilateral USRT is?
Solution:
Triangles QRS, QSP, and PQR have the same height and their bases are related by RS = PS = PR/2.
Hence, Area of QRS = Area of QSP = (Area of PQR)/2
Similarly, Area of PQT = Area of PRT = (Area of PQR)/2
So, Area of QRS = Area of PRT
(Area of USRT + Area of QTU) = (Area of USRT + Area of PUS)
Area of QTU = Area of PUS
Now we know, Area of QRS = Area of QSP
(Area of USRT + Area of QTU) = (Area of PUQ + Area of PUS)
Area of USRT = Area of PUQ
Hence, the required ratio is 1:1

Example 4.
Find the ratio of the area of two triangles on the same base if the ratio of their heights is 15 is to 3?
Solution:
Given that
h1 = 15
h2 = 3
we know that the Area of a triangle = ½ × base × height
Let the base of both triangles is b and the height of the first triangle is h1, the height of the second triangle is h2.
Area of 1st triangle / Area of 2nd triangle = ½ × b × h1 / ½ × b × H2 = h1/h2 = 15/3
​Hence proved
That the ratio of the area of two triangles with the same base is equal to the ratio of their heights.

Example 5.
Find the height of the triangle. If Area = 24cm² and base =4.8 cm?
Solution:
Given that
Area of triangle =24cm²
The base of a triangle (b)=4.8cm
Area of triangle = ½ × b × h
24 = ½ × 2.8 × h
24 = 1.4 × h
h = 24/1.4
h = 17.14
Therefore, the Height of a triangle is 17.14 cm.

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

1. Is the triangle on the same base and between the same parallel sides are equal in area?

Triangles on equal bases and between the same parallels are equal in area. The area of a triangle is half the product of its base and the corresponding altitude.

2. What is the relation between the area of two triangles if they have a common base and equal height?

If two triangles have the same height, then the ratio of their areas is the ratio of their bases. If 2 triangles have equal bases, then the ratio of their areas is the ratio of their height.

3. What is the ratio of the areas of two bases triangles?

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

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