Prove that area of a trapezium=sum of the parallel sides × height
Answers
Answer:
side =7height)=9
7×9=63
Answer:
Step-by-step explanation:
A trapezium, also known as a trapezoid, is a quadrilateral in which a pair of sides are parallel, but the other pair of opposite sides are non-parallel. The area of a trapezium is computed with the following formula:
\text{Area}=\frac {1}{2} × \text {Sum of parallel sides} × \text{Distance between them}.
Area= 21
×Sum of parallel sides×Distance between them.
The parallel sides are called the bases of the trapezium. Let b_1b
1
and b_2b 2
be the lengths of these bases. The distance between the bases is called the height of the trapezium. Let hh be this height. Then this formula becomes:
\text{Area}=\frac{1}{2}(b_1+b_2)h
Area= 21(b 1 +b 2)h
Given a trapezium, let b_1b
1
and b_2b
2
be the lengths of the bases, and let hh be the height. Draw a segment parallel to the bases that is halfway between the bases. This divides the trapezium into two trapeziums, each with the same height of \frac{1}{2}h.
21h.
Labeling the angles of these trapeziums:
Note the following congruences and identities due to the fact that the bases are parallel:
\begin{aligned} m \angle 4 + m \angle 5 &= 180^\circ \\ m \angle 1 + m \angle 7 &= 180^\circ \\ \angle 2 &\cong \angle 6 \\ \angle 3 &\cong \angle 8 \end{aligned}
m∠4+m∠5=180°
m∠1+m∠7= 180°
∠2≅∠6
∠3≅∠8
Now rotate the top trapezoid and place it adjacent to the bottom trapezoid in the following way:
Due to the aforementioned congruences and identities, this shape is a parallelogram. The length of its base is (b_1+b_2),(b 1 +b 2 ), and its height is \frac{1}{2}h. 21
h. This parallelogram has the same area as the trapezoid, so the area of the trapezoid is
\text{Area}=\frac{1}{2}(b_1+b_2)h.\ _\square
Area= 21 (b 1 +b 2 )h.