Advertisements
Advertisements
प्रश्न
Diagonals of rhombus ABCD intersect each other at point O.
Prove that: OA2 + OC2 = 2AD2 - `"BD"^2/2`
Advertisements
उत्तर
Diagonals of the rhombus are perpendicular to each other.
In quadrilateral ABCD, ∠AOD = ∠COD = 90°.
So, ΔAOD and ΔCOD are right-angled triangles.
In ΔAOD using Pythagoras theorem,
AD2 = OA2 + OD2
⇒ OA2 = AD2 - OD2 ....(i)
In ΔCOD using Pythagoras theorem,
CD2 = OC2 + OD2
⇒ OC2 = CD2 - OD2 ....(ii)
LHS = OA2 + OC2
= AD2 - OD2 + CD2 - OD2 ...[ From(i) and (ii) ]
= AD2 + CD2 - 2OD2
= AD2 + AD2 - 2`("BD"/2)^2` ...[ AD = CD and OD = `"BD"/2`]
= 2AD2 - `("BD")^2/2`
= RHS.
APPEARS IN
संबंधित प्रश्न
For finding AB and BC with the help of information given in the figure, complete following activity.
AB = BC .......... 
∴ ∠BAC =
∴ AB = BC = × AC
= × `sqrt8`
= × `2sqrt2`
=
In the given figure, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, find (1) EG (2) FD and (3) EF
Walls of two buildings on either side of a street are parallel to each other. A ladder 5.8 m long is placed on the street such that its top just reaches the window of a building at the height of 4 m. On turning the ladder over to the other side of the street, its top touches the window of the other building at a height 4.2 m. Find the width of the street.
In ∆PQR, point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.
In ∆ABC, seg AD ⊥ seg BC, DB = 3CD.
Prove that: 2AB2 = 2AC2 + BC2
AD is drawn perpendicular to base BC of an equilateral triangle ABC. Given BC = 10 cm, find the length of AD, correct to 1 place of decimal.
In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.
Prove that : 2AC2 = 2AB2 + BC2
The sides of a certain triangle is given below. Find, which of them is right-triangle
6 m, 9 m, and 13 m
In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm
Find the Pythagorean triplet from among the following set of numbers.
2, 6, 7
Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.
In an equilateral triangle ABC, the side BC is trisected at D. Prove that 9 AD2 = 7 AB2.
Find the unknown side in the following triangles
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
In the adjoining figure, a tangent is drawn to a circle of radius 4 cm and centre C, at the point S. Find the length of the tangent ST, if CT = 10 cm.
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
The longest side of a right angled triangle is called its ______.
Two rectangles are congruent, if they have same ______ and ______.
Two circles having same circumference are congruent.
Points A and B are on the opposite edges of a pond as shown in the following figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
