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
Side of a triangle is given, determine it is a right triangle.
`(2a – 1) cm, 2\sqrt { 2a } cm, and (2a + 1) cm`
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.
The angle B is:
Which of the following can be the sides of a right triangle?
2 cm, 2 cm, 5 cm
In the case of right-angled triangles, identify the right angles.
Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.
Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.
In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.
In a rectangle ABCD,
prove that: AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ2 + BP2) = (AB2 + PQ2)
A ladder, 6.5 m long, rests against a vertical wall. If the foot of the ladder is 2.5 m from the foot of the wall, find up to how much height does the ladder reach?
The sides of the triangle are given below. Find out which one is the right-angled triangle?
11, 12, 15
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 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.
In triangle ABC, line I, is a perpendicular bisector of BC.
If BC = 12 cm, SM = 8 cm, find CS
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.
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.
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
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 ______.
Points A and B are on the opposite edges of a pond as shown in the following figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
