in triangle ABC P, Q AND R ARE the mid points of side AB, BC, CA.prove that
area of parallelogram APQR=( 1/2)* area of triangle[ABC]
Join PR
By the mid-point of triangle theorem,
PR is || to BC and PR = (1/2)BC
Also triange APR is similar to triange ABC
so the areas are proportional to the square of their sides.
since the sides are 1:2
their areas are 1:4
so triangle APR = (1/4) of triangle ABC
Similary BQP would be 1/4 of triangle ABC, and
RQC is 1/4 of triangle ABC, leaving the inside triangle PQR also as 1/4 of triangle ABC to get 4/4
so figure APQR = 2/4 or 1/2 of triangle ABC
How traingle ABC and APR are similar
Since BC and PR are similar
angle B = angle APR
and angle A is common
if 2 angles of a triangle are equal to 2 corresponding angles of another triangle, the triangles are similar.
Same argument for the other pairs of similar triangles.
To prove that the area of parallelogram APQR is half the area of triangle ABC, we need to show that the two figures have equal areas.
Here's how you can prove it:
1. Recall that the area of a triangle is given by the formula: Area = (base * height) / 2. Let's denote the area of triangle ABC as A.
2. Let's find the area of parallelogram APQR. Since APQR is a parallelogram, the area can be found by considering the base and height of the parallelogram.
3. Note that side AB is parallel to side PR, so the length of PR is equal to the length of AB. Similarly, side BC is parallel to side AQ, so the length of AQ is equal to the length of BC.
4. As P, Q, and R are the midpoints of sides AB, BC, and CA respectively, we can say that PR is parallel to BC, and AQ is parallel to AB.
5. Since PR is parallel to BC, they have the same length. Similarly, AQ is parallel to AB, so they have the same length as well.
6. Now, let's consider the base and height of the parallelogram APQR. The base is PR, which has the same length as BC. The height is AQ, which has the same length as AB.
7. Thus, the base and height of parallelogram APQR are equal to the base and height of triangle ABC, respectively.
8. Using the formula for the area of a parallelogram (Area = base * height), we can write the area of parallelogram APQR as (BC * AB).
9. Since BC and AB are the sides of triangle ABC, the area of parallelogram APQR can be written as A, which is the area of triangle ABC.
10. Therefore, the area of parallelogram APQR is equal to the area of triangle ABC (Area of APQR = A = Area of ABC).
11. Finally, we can conclude that the area of parallelogram APQR is half the area of triangle ABC, as required to be proven.