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
A man goes 10 m due east and then 24 m due north. Find the distance from the starting point
Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.
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`
=
Some question and their alternative answer are given. Select the correct alternative.
If a, b, and c are sides of a triangle and a2 + b2 = c2, name the type of triangle.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
In the following figure, OP, OQ, and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC.
Prove that: AR2 + BP2 + CQ2 = AQ2 + CP2 + BR2
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'.
In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.
Find the Pythagorean triplet from among the following set of numbers.
4, 5, 6
A ladder 15m long reaches a window which is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street.
In a right-angled triangle ABC,ABC = 90°, AC = 10 cm, BC = 6 cm and BC produced to D such CD = 9 cm. Find the length of AD.
If ‘l‘ and ‘m’ are the legs and ‘n’ is the hypotenuse of a right angled triangle then, l2 = ________
In ΔABC, if DE || BC, AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, then value of x is ______.
The perimeters of two similar triangles ABC and PQR are 60 cm and 36 cm respectively. If PQ = 9 cm, then AB equals ______.
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 ______.
In a quadrilateral ABCD, ∠A + ∠D = 90°. Prove that AC2 + BD2 = AD2 + BC2
[Hint: Produce AB and DC to meet at E.]
If two legs of a right triangle are equal to two legs of another right triangle, then the right triangles are congruent.
Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?
