Advertisements
Advertisements
प्रश्न
In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ2 = 4PM2 – 3PR2.
Advertisements
उत्तर
Given: M is the midpoint of QR. ∠PRQ = 90°.
To prove: PQ2 = 4PM2 – 3PR2
Proof:
In ∆PRM,
∠PRM = 90° ...(Given)
By Pythagoras theorem,
∴ PM2 = PR2 + RM2
∴ RM2 = PM2 − PR2 ...(1)
In ∆PRQ,
∠PRQ = 90° ...(Given)
By Pythagoras theorem,
∴ PQ2 = PR2 + RQ2
∴ PQ2 = PR2 + (RM + MQ)2 ...[M is the midpoint of QR]
∴ PQ2 = PR2 + (RM + RM)2
∴ PQ2 = PR2 + (2RM)2
∴ PQ2 = PR2 + 4RM2
∴ PQ2 = PR2 + 4(PM2 − PR2) ...(from 1)
∴ PQ2 = PR2 + 4PM2 − 4PR2
∴ PQ2 = 4PM2 − 3PR2
Hence, PQ2 = 4PM2 – 3PR2.
संबंधित प्रश्न
Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.
ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, ho much string does she have out (see Figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?
Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides.
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
In the given figure, point T is in the interior of rectangle PQRS, Prove that, TS2 + TQ2 = TP2 + TR2 (As shown in the figure, draw seg AB || side SR and A-T-B)
In right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.
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 given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm;
find the length of side BC.
Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m;
find the distance between their tips.
In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that CD2 = BD × AD
Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
Triangle ABC is right-angled at vertex A. Calculate the length of BC, if AB = 18 cm and AC = 24 cm.
Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.
The sides of a certain triangle is given below. Find, which of them is right-triangle
16 cm, 20 cm, and 12 cm
Show that the triangle ABC is a right-angled triangle; if: AB = 9 cm, BC = 40 cm and AC = 41 cm
Find the Pythagorean triplet from among the following set of numbers.
2, 6, 7
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
