Advertisements
Advertisements
प्रश्न
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
Advertisements
उत्तर
Given: PS is the altitude of ΔPQR.
In ΔPSQ and ΔPSR,
∠PSQ ≅ ∠PSR ......[Each angle is equal to 90°]
PS ≅ SP ......[Common side]
PQ ≅ PR ......[Sides of an equilateral triangle]
By R.H.S. criterion of congruence,
ΔPSQ ≅ ΔPSR
∴ QS ≅ SR ......[C.S.C.T.]
Now, QS + SR = QR
QS + QS = QR .......[∵ SR = QS]
2QS = QR
QS = `(QR)/2` ......(i)
In right-angled triangle PQS, by Pythagoras theorem,
PS2 + QS2 = PQ2
PS2 + QS2 = QR2 ......[∵ PQ = QR]
PS2 = QR2 – QS2
ps2 = (2QS)2 – QS2 ......[∵ QR = 2QS]
PS2 = 4QS2 – QS2
PS2 = 3QS2
Hence proved.
APPEARS IN
संबंधित प्रश्न
Two towers of heights 10 m and 30 m stand on a plane ground. If the distance between their feet is 15 m, find the distance between their tops
If ABC is an equilateral triangle of side a, prove that its altitude = ` \frac { \sqrt { 3 } }{ 2 } a`
From a point O in the interior of a ∆ABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove
that :
`(i) AF^2 + BD^2 + CE^2 = OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2`
`(ii) AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2`
ABC is an equilateral triangle of side 2a. Find each of its altitudes.
An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after `1 1/2` hours?
In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
`"AC"^2 = "AD"^2 + "BC"."DM" + (("BC")/2)^2`
Identify, with reason, if the following is a Pythagorean triplet.
(10, 24, 27)
In the given figure, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, find (1) EG (2) FD and (3) EF
Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.
In ∆ABC, seg AD ⊥ seg BC, DB = 3CD.
Prove that: 2AB2 = 2AC2 + BC2
Diagonals of rhombus ABCD intersect each other at point O.
Prove that: OA2 + OC2 = 2AD2 - `"BD"^2/2`
Find the side of the square whose diagonal is `16sqrt(2)` cm.
In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm
Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 = AD2 - BC x CE + `(1)/(4)"BC"^2`
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 + AC2 = 2AD2 + `(1)/(2)"BC"^2`
PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.
Choose the correct alternative:
If length of sides of a triangle are a, b, c and a2 + b2 = c2, then which type of triangle it is?
In figure, PQR is a right triangle right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, find QS, RS and QR.
The top of a broken tree touches the ground at a distance of 12 m from its base. If the tree is broken at a height of 5 m from the ground then the actual height of the tree is ______.
