prove that the diagonals of isosceles trapezium are equal
To prove that the diagonals of an isosceles trapezium are equal, we will use the properties of isosceles trapezium and basic geometry concepts. An isosceles trapezium is a trapezium with both pairs of opposite sides equal in length.
Let's label our isosceles trapezium as ABCD, where AB and CD are the parallel sides, while AD and BC are the non-parallel sides.
To prove that the diagonals AC and BD are equal, we will use the concept of midsegment. The midsegment of a trapezium is a line segment connecting the midpoints of its legs (non-parallel sides).
Here's how we can prove it step by step:
Step 1: Draw the midsegment EF parallel to the bases AB and CD, connecting the midpoints of AD and BC.
Step 2: Since ABCD is an isosceles trapezium, the lengths of sides AD and BC are equal. Therefore, the distance from A to the midpoint of AD (point E) is equal to the distance from B to the midpoint of BC (point F).
Step 3: Similarly, since ABCD is an isosceles trapezium, the lengths of sides AB and CD are equal. Therefore, the distance from A to the midpoint of AB (point P) is equal to the distance from B to the midpoint of CD (point Q).
Step 4: Now, we have two pairs of equal sides: AE = BF, and AP = BQ.
Step 5: By the triangle midsegment theorem, the length of a midsegment is equal to half the sum of the lengths of the two sides it connects. Therefore, we can write:
AP + BQ = AE + EF + BF
Step 6: Since AE = BF (as mentioned in Step 4) and AP = BQ (as AB = CD), we can write:
2AP = 2BQ
Step 7: Dividing both sides of the equation by 2, we get:
AP = BQ
Step 8: From Step 3, we know that AP is the length of diagonal AC and BQ is the length of diagonal BD. Therefore:
AC = BD
Hence, we have proved that the diagonals of an isosceles trapezium are equal.