What is Isosceles Trapezium
Its a trapezium in which one pair of sides are parallel and other non parallel sides are equal.
Keyword for Isosceles Trapezium
(a) One pair of sides are parallel
(b) Non parallel sides are equal
Example of Isosceles Trapezium
Given above is the example of Isosceles Trapezium.
⟹ AB & CD are parallel sides called Base
⟹ Non Parallel sides AD & BC are equal (AD = BC)
Non Parallel sides are also called legs
Properties of Isosceles Trapezium
(01) In Isosceles Trapezium, the non parallel sides are equal
This property separates the Isosceles Trapezium from general Trapezium because in general trapezium all sides are unequal
In the above trapezium ABCD, the non parallel sides AD & BC are equal
AD = BC
(02) In Isosceles Trapezium, both adjacent and opposite angles are supplementary
Adjacent angles are supplementary
Similar to normal trapezium, the adjacent angles are supplementary in Isosceles Trapezium.
∠ 1 + ∠ 2 = 180 degree
∠ 3 + ∠ 4 = 180 degree
Opposite angles in Isosceles Trapezium are also supplementary
This property is special for Isosceles Trapezium
∠ 1 + ∠ 4 = 180 degree
∠ 3 + ∠ 2 = 180 degree
(03) In Isosceles Trapezium, the diagonals are equal to each other
ABCD is an Isosceles Trapezium in which AC & BD are diagonals.
According to diagonal property of Isosceles Trapezium
AC = BD
(04) The diagonals divides into equal proportion
In trapezium, diagonals intersect such that:
\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}
Also in case of Isosceles Trapezium
DO = OC and;
OB = OC
(05) Base angles are equal
Given above is the isosceles trapezium ABCD in which:
∠ ADC = ∠ BCD
Prove Base angles in Isosceles Trapezium are equal
Given
ABCD is an isosceles trapezium in which AB II CD and side AD = BC
To prove
∠ ADC = ∠ BCD
Solution
Draw diagonals AC & BD in the trapezium
Now take triangle ▵ DAC & ▵ CBD
(Separated the triangles for your understanding)
Side DA = CB (Legs of Isosceles Trapezium are equal)
AC = BD (Diagonals in Isosceles Trapezium are equal)
DC = DC (Common Side)
Hence by SSS congruency, ▵ DAC & ▵ CBD are congruent
so, ∠ ADC = ∠ BCD
Hence Proved
(06) Area of Isosceles Trapezium
Area = (1/2) x (Sum of Parallel sides ) x height
Area = (1/2) x (AB + CD) x AO
(07) Converting Isosceles Trapezium into Rectangle
Given above is an Isosceles Trapezium ABCD.
Here we have plotted lines AM & BN as height of trapezium perpendicular to line DC
Comparing triangle ADM & BNC
AD = BC {equal sides
AM = BN {similar height}
∠ ADM = ∠ BCN
On comparison, we can say that:
DM = CN
Now cut the triangle AMD from dotted line and paste above triangle BCN as shown in below figure
Observe how we have transformed a trapezium into a rectangle.
This method is very helpful to solve problems related to area calculation
Example Question
If ABCD is an Isosceles Trapezium in which diagonal length AC = 10 cm and height 5 cm. Find the area of Trapezium
Now as we have seen in above concept, remove the triangle AOD & attach towards side BC
After all the deliberations, we got rectangle ADCO with breadth 5 cm and diagonal 10 cm
Using Pythagoras Theorem to find length of OC
\mathsf{AC^{2} =\ OA^{2} +\ OC^{2}}\\\ \\ \mathsf{10^{2} =\ 5^{2} +\ OC^{2}}\\\ \\ \mathsf{OC^{2} \ =\ 100\ -25}\\\ \\ \mathsf{OC^{2} \ =\ 75}\\\ \\ \mathsf{OC\ =\ 8.6\ cm} \\ \\
We know,
Area of Rectangle = OA x OC = 5 x 8.6 = 43 sq cm
Hence, 43 sq cm is the original area of trapezium
(08) Isosceles Trapezium’s Circumcircle
Isosceles Trapezium can be fully inscribed in circumcircle.
Circumcircle is a circle which inscribe the geometrical figure by passing through all the vertex.
Location of Circumcenter
Circumcenter location can be found by the intersection of perpendicular bisector of adjacent sides
Given above is Isosceles Trapezium ABCD
M & N are perpendicular bisectors of side AD & CD.
The bisectors meet at point O, which is circumcenter of the trapezium.
Take OC as radius and draw a circle.
Note that circle passes through all the vertex
