Home
Class 11
MATHS
Let A B C D be a rectangle and P be any ...

Let `A B C D` be a rectangle and `P` be any point in its plane. Show that `A P^2+P C^2=P B^2+P D^2dot`

Text Solution

AI Generated Solution

To show that \( AP^2 + PC^2 = PB^2 + PD^2 \) for a rectangle \( ABCD \) and a point \( P \) in its plane, we will use coordinate geometry. Let's assume the rectangle \( ABCD \) is positioned in the Cartesian plane as follows: - \( A(0, 0) \) - \( B(a, 0) \) - \( C(a, b) \) - \( D(0, b) \) Let \( P \) be any point in the plane with coordinates \( P(x, y) \). ...
Promotional Banner

Similar Questions

Explore conceptually related problems

In Figure, A B C D\ a n d\ P Q R C are rectangles and Q is the mid-point of A C . Prove that D P=P C (ii) P R=1/2\ A C

The coordinates of A ,B ,C are (6,3),(-3,5),(4,-2) , respectively, and P is any point (x ,y) . Show that the ratio of the area of P B C to that of A B C is (|x+y-2|)/7dot

In A B C ,D is the mid-point of A B ,P is any point of B CdotC Q P D meets A B in Q . Show that a r( B P Q)=1/2a r( A B C)dot TO PROVE : a r( B P Q)=1/2a r( A B C) CONSTRUCTION : Join CD.

Let a,b,c,d and p be any non zero distinct real numbers such that (a^(2) + b^(2) + c^(2))p^(2) - 2(ab+bc+cd)p+ (b^(2) + c^(2) + d^(2))=0 . Then :

If the angles A,B,C of /_\ABC are in A.P., and its sides a, b, c are in Gp., show that a^(2) , b^(2), c^(2) are in A.P.

A B C D is a parallelogram. P is a point on A D such that A P=1/3\ A D\ a n d\ Q is a point on B C such that: C Q=1/3B Cdot Prove that A Q C P is a parallelogram.

In Figure, P is a point in the interior of a parallelogram A B C D . Show that a r( A P B)+a r( P C D)=1/2a r(^(gm)A B C D) a R(A P D)+a r( P B C)=a r( A P B)+a r( P C D)

p A N D q are any two points lying on the sides D C a n d A D respectively of a parallelogram A B C Ddot Show that a r( A P B)=a r ( B Q C)dot

In Fig. 9.32, ABCD is a parallelogram and BC is produced to a point Q such that A D" "=" "C Q . If AQ intersect DC at P, show that a r" "(B P C)" "=" "a r" "(D P Q)dot

If a,b,c and d are in G.P , then show that ax^(2)+c divides ax^(3)+bx^(2)+cx+d