Advertisements
Advertisements
प्रश्न
In ∆ABC, ∠BAC = 90°, seg BL and seg CM are medians of ∆ABC. Then prove that:
4(BL2 + CM2) = 5 BC2
Advertisements
उत्तर
Given: Δ ABC right angled at A, i.e., A = 90, where BL and CM are the medians.
To Prove: 4(BL2 + CM2) = 5BC2
Proof:
Since BL is the median,
AL = CL = `1/2` AC ...(1)
Similarly, CM is the median
AM = MB = `1/2`AB ...(2)
∴ by Pythagoras theorem,
(Hypotenuse)2 = (Height)2 + (Base)2 ...(3)
In ΔBAC,
(BC)2 = (AB)2 + (AC)2 ...(4)
∠BAC = 90°
In ΔBAL,
(BL)2 = AB2 + AL2 ...(From 1)
BL2 = AB2 + `(("AC")/2)^2`
BL2 = AB2 + `("AC"^2/4)`
Multiply both sides by 4,
4BL2 = 4AB2 + AC2 ...(5)
In ΔMAC,
CM2 = AM2 + AC2 ...(From 2)
CM2 = `(("AB")/2)^2` + AC2
CM2 = `("AB")^2/4` + AC2
Multiply both sides by 4,
4CM2 = AB2 + 4AC2 ...(6)
Adding 5 and 6,
4BL2 + 4CM2 = (4AB2 + AC2) + (AB2 + 4AC2)
4(BL2 + CM2) = 5AB2 + 5AC2
∴ 4(BL2 + CM2) = 5BC2
Hence, proved.
APPEARS IN
संबंधित प्रश्न
If ABC is an equilateral triangle of side a, prove that its altitude = ` \frac { \sqrt { 3 } }{ 2 } a`
A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD2 = 7 AB2
In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.
In ∆ABC, seg AD ⊥ seg BC, DB = 3CD.
Prove that: 2AB2 = 2AC2 + BC2
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.
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 triangle ABC, AB = AC and BD is perpendicular to AC.
Prove that: BD2 − CD2 = 2CD × AD
O is any point inside a rectangle ABCD.
Prove that: OB2 + OD2 = OC2 + OA2.
Use the information given in the figure to find the length AD.
The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder?
Find the Pythagorean triplet from among the following set of numbers.
4, 5, 6
From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.
Find the length of the hypotenuse of a triangle whose other two sides are 24cm and 7cm.
A point OI in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D. Prove that OB2 + OD2 = OC2 + OA2
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
If in a ΔPQR, PR2 = PQ2 + QR2, then the right angle of ∆PQR is at the vertex ________
Choose the correct alternative:
If length of sides of a triangle are a, b, c and a2 + b2 = c2, then which type of triangle it is?
Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.
Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.
