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
If ABC is an equilateral triangle of side a, prove that its altitude = ` \frac { \sqrt { 3 } }{ 2 } a`
In a right triangle ABC right-angled at C, P and Q are the points on the sides CA and CB respectively, which divide these sides in the ratio 2 : 1. Prove that
`(i) 9 AQ^2 = 9 AC^2 + 4 BC^2`
`(ii) 9 BP^2 = 9 BC^2 + 4 AC^2`
`(iii) 9 (AQ^2 + BP^2 ) = 13 AB^2`
In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
(i) OA2 + OB2 + OC2 − OD2 − OE2 − OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2
Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides.
Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.
In the given figure, ∠B = 90°, XY || BC, AB = 12 cm, AY = 8cm and AX : XB = 1 : 2 = AY : YC.
Find the lengths of AC and BC.
In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.
Prove that : 2AC2 = 2AB2 + BC2
In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.
Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.
In the given figure, angle ACP = ∠BDP = 90°, AC = 12 m, BD = 9 m and PA= PB = 15 m. Find:
(i) CP
(ii) PD
(iii) CD
In the given figure, AD = 13 cm, BC = 12 cm, AB = 3 cm and angle ACD = angle ABC = 90°. Find the length of DC.
A right triangle has hypotenuse p cm and one side q cm. If p - q = 1, find the length of third side of the triangle.
The length of the diagonals of rhombus are 24cm and 10cm. Find each side of the rhombus.
In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that: 9BP2 = 9BC2 + 4AC2
Find the unknown side in the following triangles
Find the distance between the helicopter and the ship
Rithika buys an LED TV which has a 25 inches screen. If its height is 7 inches, how wide is the screen? Her TV cabinet is 20 inches wide. Will the TV fit into the cabinet? Give reason
Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.
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.
In a right-angled triangle ABC, if angle B = 90°, BC = 3 cm and AC = 5 cm, then the length of side AB is ______.
