Students who are in search of the theorems on Area can make use of our page. Here you can find the statements of the theorems with proofs in a very simple manner. Click on the relevant links and learn the theorems and apply them to the questions so that you can solve the problems easily. In the previous articles, we have discussed proving Triangles on the Same Base and between the Same Parallels are Equal in Area. Now on this page, we will prove that Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes.
Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes Theorem & Proof
Theorem:
Prove that Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes?
Proof:
Given
ABC and DBC are two triangles on the same base BC, and ar(∆ABC) = ar(∆DBC). Also, AN and DM are their corresponding altitudes.
To prove
AN = DM (or AD ∥ BC).
Construction
Join AD.
Proof
1/2 × BC × AN = 1/2 × BC × DM…. equation 1
[Area of a triangle = 1/2 × base × altitude, and ar(∆ABC) = ar(∆DBC)]
PN = SM……. equation 2
Cancelling ½ × BC 1/2×BC from equation 1
AN ∥ DM.[AN ⊥ BR and SM ⊥ BC]…. equation 3
ANMD is a rectangle……. equation 4[ AMND is a parallelogram]
By equations 2 and 3, two angles are right angles.
AN = DM or (AD ∥ BC)…. equation 5
Hence proved
By equation 4, ANMD is a rectangle.
Do Check:
- Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels
- Triangles on the Same Base and between the Same Parallels are Equal in Area
FAQs on Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes
1. Do all triangles with the same bases and heights have the same areas?
Triangles on the same base or 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. Triangles on the same base or equal bases and having equal areas lie in between the same parallels. A median of a triangle divides the two triangles into equal areas.
2. Do triangles have the same area?
Triangles the ratio of their areas is equal to the square of the ratio of their pair of corresponding sides. So, that the areas of two triangles cannot be equal. But the congruent triangles always have equal areas.
3. Is it possible for a triangle to have the same area and perimeter?
Triangles with the same perimeter and same area and with one side the same are congruent. The area of a triangle is half of the product of the base and the perpendicular height of a triangle. When the given base and the area, the height is fixed.