Advertisements
Advertisements
Question
In the given figure, triangle PQR is right-angled at Q. S is the mid-point of side QR. Prove that QR2 = 4(PS2 – PQ2).
Solution
Given: In triangle PQR, ∠PQR = 90° and S is the mid-point of QR.
To prove: QR2 = 4(PS2 – PQ2)
in right-angled ΔPQS, by Pythagoras theorem,
PQ2 + QS2 = PS2
⇒ QS2 = PS2 – PQ2 .......(i)
Since S is the mid-point of side QR,
∴ QS = `(QR)/2`
Substituting the value of QS in equation (i),
`((QR)/2)^2 = PS^2 - PQ^2`
`(QR^2)/4 = PS^2 - PQ^2`
QR2 = 4(PS2 – PQ)2
Hence proved.
APPEARS IN
RELATED QUESTIONS
5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.
In the adjacent figure, ABC is a right angled triangle with right angle at B and points D, E trisect BC. Prove that 8AE2 = 3AC2 + 5AD2
If in ∆ABC, DE || BC. AB = 3.6 cm, AC = 2.4 cm and AD = 2.1 cm then the length of AE is
In a ∆ABC, AD is the bisector of ∠BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm. The length of the side AC is
Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, what is the distance between their tops?
If the sides of a triangle are in the ratio 5 : 12 : 13 then, it is ________
Check whether given sides are the sides of right-angled triangles, using Pythagoras theorem
8, 15, 17
Check whether given sides are the sides of right-angled triangles, using Pythagoras theorem
30, 40, 50
Choose the correct alternative:
In right angled triangle, if sum of the squares of the sides of right angle is 169, then what is the length of the hypotenuse?
Choose the correct alternative:
A rectangle having length of a side is 12 and length of diagonal is 20, then what is length of other side?
Choose the correct alternative:
In ∆ABC, AB = `6sqrt(3)` cm, AC = 12 cm, and BC = 6 cm, then m∠A = ?
If a triangle having sides 8 cm, 15 cm and 17 cm, then state whether given triangle is right angled triangle or not
A rectangle having dimensions 35 m × 12 m, then what is the length of its diagonal?
In ∆LMN, l = 5, m = 13, n = 12 then complete the activity to show that whether the given triangle is right angled triangle or not.
*(l, m, n are opposite sides of ∠L, ∠M, ∠N respectively)
Activity: In ∆LMN, l = 5, m = 13, n = `square`
∴ l2 = `square`, m2 = 169, n2 = 144.
∴ l2 + n2 = 25 + 144 = `square`
∴ `square` + l2 = m2
∴By Converse of Pythagoras theorem, ∆LMN is right angled triangle.
In ΔABC, AB = 9 cm, BC = 40 cm, AC = 41 cm. State whether ΔABC is a right-angled triangle or not. Write reason.
In a right angled triangle, right-angled at B, lengths of sides AB and AC are 5 cm and 13 cm, respectively. What will be the length of side BC?
