Advertisements
Advertisements
Question
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
Advertisements
Solution
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
RELATED QUESTIONS
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
(A)\[7 + \sqrt{5}\]
(B) 5
(C) 10
(D) 12
Prove that the points A(0, −1), B(−2, 3), C(6, 7) and D(8, 3) are the vertices of a rectangle ABCD?
In the figure: ∠PSQ = 90o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.
The given figure shows a quadrilateral ABCD in which AD = 13 cm, DC = 12 cm, BC = 3 cm and ∠ABD = ∠BCD = 90o. Calculate the length of AB.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
Find the length of diagonal of the square whose side is 8 cm.
Prove that `(sin θ + cosec θ)^2 + (cos θ + sec θ)^2 = 7 + tan^2 θ + cot^2 θ`.
Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.
Show that the triangle ABC is a right-angled triangle; if: AB = 9 cm, BC = 40 cm and AC = 41 cm
In the figure below, find the value of 'x'.
Find the Pythagorean triplet from among the following set of numbers.
3, 4, 5
Find the Pythagorean triplet from among the following set of numbers.
9, 40, 41
From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.
PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.
If in a ΔPQR, PR2 = PQ2 + QR2, then the right angle of ∆PQR is at the vertex ________
From the given figure, in ∆ABQ, if AQ = 8 cm, then AB = ?
In a right angled triangle, if length of hypotenuse is 25 cm and height is 7 cm, then what is the length of its base?
Sides AB and BE of a right triangle, right-angled at B are of lengths 16 cm and 8 cm respectively. The length of the side of largest square FDGB that can be inscribed in the triangle ABE is ______.
