Advertisements
Advertisements
Question
The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2
Advertisements
Solution
Given QS = 3SR
QR = QS + SR
= 3SR + SR = 4SR
SR = `1/4` QR ...(1)
QS = 3SR
SR = `"QS"/3` ...(2)
From (1) and (2) we get
`1/4 "QR" = "QS"/3`
∴ QS = `3/4` QR ...(3)
In the right ∆PQS,
PQ2 = PS2 + QS2 ...(4)
Similarly in ∆PSR
PR2 = PS2 + SR2 ...(5)
Subtract (4) and (5)
PQ2 – PR2 = PS2 + QS2 – PS2 – SR2
= QS2 – SR2
PQ2 – PR2 = `[3/4 "QR"]^2 - ["QR"/4]^2`
From (3) and (1)
= `(9"QR"^2)/16 - "QR"^2/16`
= `(8"QR"^2)/16`
PQ2 – PR2 = `1/2 "QR"^2`
2PQ2 – 2PR2 = QR2
2PQ2 = 2PR2 + QR2
Hence the proved.
APPEARS IN
RELATED QUESTIONS
In a ∆ABC, AD ⊥ BC and AD2 = BC × CD. Prove ∆ABC is a right triangle
Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was \[15\sqrt{2}\]
km. Find their speed per hour.
In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.
In the figure below, find the value of 'x'.
Find the Pythagorean triplet from among the following set of numbers.
3, 4, 5
The sides of the triangle are given below. Find out which one is the right-angled triangle?
1.5, 1.6, 1.7
Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.
Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?
The top of a broken tree touches the ground at a distance of 12 m from its base. If the tree is broken at a height of 5 m from the ground then the actual height of the tree is ______.
Two trees 7 m and 4 m high stand upright on a ground. If their bases (roots) are 4 m apart, then the distance between their tops is ______.
