There are some conditions in solving the theorem Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels. We can prove that the Area of a Triangle is Half that of a Parallelogram only when the bases of the triangle and parallelogram have the same base. If the bases are the same then we can say that the area of the triangle is half that of the parallelogram. Go through the below section to know how to prove the given statement.
Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels
Prove that the Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels?
PQRS is a parallelogram and PQM is a triangle with the same base PQ, and is between the same parallel lines PQ and SR.
To prove:
ar(∆PQM) = 12 × ar(Parallelogram PQRS).
Construction:
Draw MN ∥ SP which cuts PQ at N.
Proof:
SM ∥ PN [SR ∥ PQ being opposite sides of the parallelogram PQRS.]…. equation 1
SP ∥ MN [ By construction ]… equation 2
PNMS is a parallelogram [By definition of parallelogram because of equations 1 and 2.]
ar(∆PNM) = ar(∆PSM) [ PM is a diagonal of the parallelogram PNMS ]…. equation 4
2ar(∆PNM) = ar(∆PSM) + ar(∆PNM) [Adding the same area on both sides of equality in equation 4.]…. equation 5
2ar(∆PNM) = ar(parallelogram PNMS) [By addition axiom of area]…. equation 6
MN ∥ RQ [A line parallel to one of the two parallel lines, is also parallel to the other line]…… equation 7
MNQR is a parallelogram.[Similar to equation 3]…… equation 8
2ar(∆MNQ) = ar(parallelogram MNQR) [ Similar to equation 6]…… equation 9
2{ar(∆PNM) + ar(∆MNQ)} = ar(parallelogram PNMS) + ar(parallelogram MNQR) [Adding equation 6 and 9]……. equation 10
2ar(∆PQM) = ar(parallelogram PQRS), that is ar(∆PQM) = 1/2 × ar(parallelogram PQRS). [By addition axiom of area]…… equation 11.
Hence proved.
Corollaries:
- Area of the triangle = 1/2 × base × height (altitude)
- If a triangle and a parallelogram have equal bases and are between the same parallels then ar(triangle) = 1/2 × ar(parallelogram)
See More:
- Area and Perimeter of Triangle
- Every Diagonal of a Parallelogram Divides it into Two Triangles of Equal Area
- Triangles on the Same Base and between the Same Parallels are Equal in Area
FAQs on Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels
1. Why is the area of a triangle half the area of a parallelogram?
The area of a parallelogram is measured the same as the area for a rectangle and a square. We can divide a square or a quadrilateral into two triangles. It gives us the area of a triangle is half the area of the quadrilateral with the same base and height.
2. Does a triangle have parallel sides?
A triangle is a geometric shape it has three sides and three angles. Triangles have no parallel lines or perpendicular lines.
3. What do a parallelogram and a triangle have in common?
Triangles have all the properties of a parallelogram. Parallelograms have opposite sides that are parallel, their diagonals bisect each other and divide the parallelogram into two congruent triangles, and opposite sides and angles of a parallelogram are congruent.