Name: Problem Based on Midpoint Theorem
Uploaded: 2019-07-19
Duration: 5 min 16 s
Description: The line segment in a triangle joining the midpoint of two sides of the triangle is parallel to its third side and is also half of the length of the third side.

Question

A B C D

is a kite having

A B = A D

and

B C = C D

. Prove that the figure formed by joining the midpoints of the sides, in order, is a rectangle.

Accepted Answer

Let us consider the given figure as:

Given, $A B = A D$ is a kite with $B C = C D$ and $B C = C D$ .

$E, F, G, H$ are the midpoints of sides $A B, B C, C D$ and $D A$ respectively. By joining these points in order, we get a quadrilateral $E F G H$ .

Construction: We shall join the points $B$ and $D$ , $A$ and $C$ .

Proof: In $△ A B D, E$ and $H$ are midpoints of $A B$ and $A D$ respectively.

Thus, by midpoint theorem, we get,

$E H = \frac{1}{2} B D$ ........(i)

And $E H | | B D$ ........(ii)

Again, in $△ C B D$ , $F$ and $G$ are midpoints of $B C$ and $C D$ respectively.

Thus, by midpoints theorem, we get,

$F G = \frac{1}{2} B D$ ........(iii)

By (i) and (iii) we get,

$E H = F G$ .......(iv)

Similarly, from $△ A B C$ and $△ A C D$ , by applying mid-point theorem, we can prove,

$E F = H G$ .......(v)

We know, in a kite, the diagonals intersect at right angles.

Thus, we get $∠ P Q R = 90 °$

Now, from the quadrilateral $P Q R E$ , we get,

$E R | | P Q$ [By (ii)]

$P E | | Q R$ [By applying midpoint theorem on $△ A B C$ ]

So, the quadrilateral $P Q R E$ is a parallelogram with one of its angle $∠ P Q R = 90 °$ .

Hence, $P Q R E$ is a rectangle.

Therefore, $∠ P E R = 90 °$ ......(vi)

Thus, from (iv), (v) and (vi), $E F G H$ is a parallelogram whose one angle, $∠ P E R = 90 °$ .

Therefore, $E F G H$ is a rectangle.

$A B C D$ is a kite having $A B = A D$ and $B C = C D$ . Prove that the figure formed by joining the midpoints of the sides, in order, is a rectangle.

Important Questions on Mid-Point and Intercept Theorems

Learn and Prepare for any exam you want !

ABCD is a kite having AB=AD and BC=CD. Prove that the figure formed by joining the midpoints of the sides, in order, is a rectangle.

Important Questions on Mid-Point and Intercept Theorems

Learn and Prepare for any exam you want !

$A B C D$ is a kite having $A B = A D$ and $B C = C D$ . Prove that the figure formed by joining the midpoints of the sides, in order, is a rectangle.