Advertisements
Advertisements
प्रश्न
In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that: 9AQ2 = 9AC2 + 4BC2
Advertisements
उत्तर
P divides AC in the ratio 2 : 1
So C.P. = `(2)/(3) "AC"` .......(i)
Q divides BC in the ratio 2 : 1
QC = `(2)/(3)"BC"` ......(ii)
In ΔACQ
Using Pythagoras Theorem we have,
AQ2 + AC2 + CQ2
⇒ AQ2 = `"AC"^2 + (4)/(9)"BC"^2` ...(using (ii))
⇒ 9AQ2 = 9AC2 + 4BC2. ......(iii)
APPEARS IN
संबंधित प्रश्न
In triangle ABC, ∠C=90°. Let BC= a, CA= b, AB= c and let 'p' be the length of the perpendicular from 'C' on AB, prove that:
1. cp = ab
2. `1/p^2=1/a^2+1/b^2`
The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right traingle ,right-angled at B. Find the values of p.
If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.
In an isosceles triangle ABC; AB = AC and D is the point on BC produced.
Prove that: AD2 = AC2 + BD.CD.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
The sides of a certain triangle is given below. Find, which of them is right-triangle
16 cm, 20 cm, and 12 cm
In the given figure, angle BAC = 90°, AC = 400 m, and AB = 300 m. Find the length of BC.
In triangle PQR, angle Q = 90°, find: PR, if PQ = 8 cm and QR = 6 cm
In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm
The sides of the triangle are given below. Find out which one is the right-angled triangle?
11, 12, 15
A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.
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 triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 = AD2 - BC x CE + `(1)/(4)"BC"^2`
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
In a square PQRS of side 5 cm, A, B, C and D are points on sides PQ, QR, RS and SP respectively such as PA = PD = RB = RC = 2 cm. Prove that ABCD is a rectangle. Also, find the area and perimeter of the rectangle.
To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
If S is a point on side PQ of a ΔPQR such that PS = QS = RS, then ______.
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 ______.
Two rectangles are congruent, if they have same ______ and ______.
