Name: Rectangles and Squares Formed by Four Points
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: What are the conditions for a quadrilateral to be a rectangle? How does it differ from a square? Unlock your doubts and clear your basics from this video!

Question

Let

A B C

be an isosceles triangle in which

A B = A C

. If

D, E, F

be the midpoints of the sides

B C, C A

and

A B

respectively, show that the segment

A D

and

E F

bisect each other at right angles.

Accepted Answer

$△ A B C,$ an isosceles triangle is given with $D, E$ and $F$ as the midpoints of $B C, C A$ and $A B$ respectively as shown below.

We need to prove that the segment $A D$ and $E F$ bisect each other at right angle.

Let's join $D F$ and $D E$ .

In $△ A B C, D$ and $E$ are the midpoints of $B C$ and $A C$ respectively.

Midpoint theorem states, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get $D E ∥ A B$ Or $D E ∥ A F$ .

Similarly, we can get $D F ∥ A E$ .

Therefore, $A E D F$ is a parallelogram.

We know that opposite sides of a parallelogram are equal.

$D F = A E$ and $D E = A F$

Also, from the midpoint theorem we get $D E = \frac{1}{2} A B$

Similarly, $D F = \frac{1}{2} A C$

It is given that $△ A B C$ is an isosceles triangle where $A B = A C$ .

$\Rightarrow \frac{1}{2} A B = \frac{1}{2} A C \Rightarrow D E = D F$

Also, $\frac{1}{2} A B = \frac{1}{2} A C = A E = A F$

So, $A E D F$ is a rhombus.

We know that the diagonals of a rhombus bisect each other at right angle.

Therefore, $M$ is the midpoint of $E F$ and $A M ⊥ E F$ .

Hence, proved.

Let $A B C$ be an isosceles triangle in which $A B = A C$ . If $D, E, F$ be the midpoints of the sides $B C, C A$ and $A B$ respectively, show that the segment $A D$ and $E F$ bisect each other at right angles.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

Let ABC be an isosceles triangle in which AB=AC. If D,E,F be the midpoints of the sides BC,CA and AB respectively, show that the segment AD and EF bisect each other at right angles.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

Let $A B C$ be an isosceles triangle in which $A B = A C$ . If $D, E, F$ be the midpoints of the sides $B C, C A$ and $A B$ respectively, show that the segment $A D$ and $E F$ bisect each other at right angles.