Converse of BPT

Q. In Figure, P is a point in the interior of a parallelogram ABCD. Show that
$\text{(i)}ar\left(APB\right)+ar\left(PCD\right)=\frac{1}{2}ar\left(ABCD\right)$ $\text{(ii)}ar\left(APD\right)+ar\left(PBC\right)=ar\left(APB\right)+ar\left(PCD\right)$ [Hint : Through P, draw a line parallel to AB.]

Q.

In $\Delta$ABC, if DE divides AB and AC in the same ratio, then which of the following options is true?

1. AD = AE

2. AD = DB

3. DE is half of BC

4. DE and BC are parallel

Q. In Figure, P is a point in the interior of a parallelogram ABCD. Show that
$\text{(i)}ar\left(APB\right)+ar\left(PCD\right)=\frac{1}{2}ar\left(ABCD\right)$ $\text{(ii)}ar\left(APD\right)+ar\left(PBC\right)=ar\left(APB\right)+ar\left(PCD\right)$ [Hint : Through P, draw a line parallel to AB.]

Q.

In the given Figure, $\frac{PS}{SQ}=\frac{PT}{TR}$ and $\angle PST=\angle PRQ.$ Prove that $PQR$ is an isosceles triangle.

Q. Question 7
P and Q are the mid – points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

Q. $ABC$ is a triangle right angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and Parallel to $BC$ intersects $AC$ at $D$. Show that
$CM=MA=\frac{1}{2}AB$

Q. Find the length of OC, if CD = 8cm and AD and BC are equal perpendiculars to line segment AB as shown below.

1. 4

Q. In given fig, $PR>PQ$ and $PS$ bisects $\angle QPR$. Prove that $\angle PSR>\angle PSQ$.

Q. Find the value of x.​