In an Equilateral Triangle, Prove that Three Times the Square of One Side is Equal to Four Times the Square of One of Its Altitudes. - Mathematics

Advertisements
Advertisements

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Advertisements

Solution

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.

`:. BE = EC = (BC)/2 = a/2`

Applying Pythagoras theorem in ΔABE, we obtain

AB2 = AE2 + BE2

`a^2 = AE^2 + (a/2)^2`

`AE^2 = a^2 - a^2/4`

`AE^2 = (3a^2)/4`

4AE2 = 3a2

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

  Is there an error in this question or solution?
Chapter 6: Triangles - Exercise 6.5 [Page 151]

APPEARS IN

NCERT Mathematics Class 10
Chapter 6 Triangles
Exercise 6.5 | Q 16 | Page 151

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

If the sides of a triangle are 6 cm, 8 cm and 10 cm, respectively, then determine whether the triangle is a right angle triangle or not.


ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is 
(A) 5
(B) 4
(C) 3
(D) 25


A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder


If ABC is an equilateral triangle of side a, prove that its altitude = ` \frac { \sqrt { 3 } }{ 2 } a`


ABCD is a rhombus. Prove that AB2 + BC2 + CD2 + DA2= AC2 + BD2


Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 50 cm, 80 cm, 100 cm

 


Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.

The angle B is:


A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.


Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.


The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.


Identify, with reason, if the following is a Pythagorean triplet.
(3, 5, 4)


Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)


Identify, with reason, if the following is a Pythagorean triplet.
(5, 12, 13)


For finding AB and BC with the help of information given in the figure, complete following activity.

AB = BC ..........

\[\therefore \angle BAC = \]

\[ \therefore AB = BC =\] \[\times AC\]

\[ =\] \[\times \sqrt{8}\]

\[ =\] \[\times 2\sqrt{2}\]

 =


Find the side and perimeter of a square whose diagonal is 10 cm ?


Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.


In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ= 4PM– 3PR2.


In ∆PQR, point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.


In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.


Digonals of parallelogram WXYZ intersect at point O. If OY =5, find WY.


A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.


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.


If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.


In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.



In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.

Prove that : 2AC2 = 2AB2 + BC2


In triangle ABC, AB = AC and BD is perpendicular to AC.
Prove that: BD2 - CD2 = 2CD × AD.


In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.


In an isosceles triangle ABC; AB = AC and D is the point on BC produced.
Prove that: AD2 = AC2 + BD.CD.


In the following figure, OP, OQ, and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC.

Prove that: AR2 + BP2 + CQ2 = AQ2 + CP2 + BR2


O is any point inside a rectangle ABCD.
Prove that: OB2 + OD2 = OC2 + OA2.


In a rectangle ABCD,
prove that: AC2 + BD2 = AB2 + BC2 + CD2 + DA2.


If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.


Choose the correct alternative: 

In right-angled triangle PQR, if hypotenuse PR = 12 and PQ = 6, then what is the measure of ∠P? 


Find the side of the square whose diagonal is `16sqrt(2)` cm.


Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.


Triangle ABC is right-angled at vertex A. Calculate the length of BC, if AB = 18 cm and AC = 24 cm.


Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.


The sides of a certain triangle is given below. Find, which of them is right-triangle

16 cm, 20 cm, and 12 cm


The sides of a certain triangle is given below. Find, which of them is right-triangle

6 m, 9 m, and 13 m


In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm


In the given figure, angle ACB = 90° = angle ACD. If AB = 10 m, BC = 6 cm and AD = 17 cm, find :
(i) AC
(ii) CD


In the given figure, angle ADB = 90°, AC = AB = 26 cm and BD = DC. If the length of AD = 24 cm; find the length of BC.


In the given figure, AD = 13 cm, BC = 12 cm, AB = 3 cm and angle ACD = angle ABC = 90°. Find the length of DC.


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?


Use the information given in the figure to find the length AD.


In the figure below, find the value of 'x'.


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.

2, 4, 5


Find the Pythagorean triplet from among the following set of numbers.

4, 5, 6


Find the Pythagorean triplet from among the following set of numbers.

2, 6, 7


The sides of the triangle are given below. Find out which one is the right-angled triangle?

8, 15, 17


The sides of the triangle are given below. Find out which one is the right-angled triangle?

11, 60, 61


The sides of the triangle are given below. Find out which one is the right-angled triangle?

1.5, 1.6, 1.7


The sides of the triangle are given below. Find out which one is the right-angled triangle?

40, 20, 30


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 ladder 25m long reaches a window of a building 20m above the ground. Determine the distance of the foot of the ladder from the building.


A right triangle has hypotenuse p cm and one side q cm. If p - q = 1, find the length of third side of the triangle.


The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall to what height does its tip reach?


In ΔABC, AD is perpendicular to BC. Prove that: AB2 + CD2 = AC2 + BD2


From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF2 + BD2 + CE= OA2 + OB2 + OC2 - OD2 - OE2 - OF2


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 


A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?


There are two paths that one can choose to go from Sarah’s house to James's house. One way is to take C street, and the other way requires to take B street and then A street. How much shorter is the direct path along C street?


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?


The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2 


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?


If in a ΔPQR, PR2 = PQ2 + QR2, then the right angle of ∆PQR is at the vertex ________


Find the unknown side in the following triangles


Find the unknown side in the following triangles


An isosceles triangle has equal sides each 13 cm and a base 24 cm in length. Find its height


Find the distance between the helicopter and the ship


In triangle ABC, line I, is a perpendicular bisector of BC.
If BC = 12 cm, SM = 8 cm, find CS


The hypotenuse of a right angled triangle of sides 12 cm and 16 cm is __________


Find the length of the support cable required to support the tower with the floor


Rithika buys an LED TV which has a 25 inches screen. If its height is 7 inches, how wide is the screen? Her TV cabinet is 20 inches wide. Will the TV fit into the cabinet? Give reason


In the figure, find AR


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?


From given figure, In ∆ABC, If AC = 12 cm. then AB =?


Activity: From given figure, In ∆ABC, ∠ABC = 90°, ∠ACB = 30°

∴ ∠BAC = `square`

∴ ∆ABC is 30° – 60° – 90° triangle

∴ In ∆ABC by property of 30° – 60° – 90° triangle.

∴ AB = `1/2` AC and `square` = `sqrt(3)/2` AC

∴ `square` = `1/2 xx 12` and BC = `sqrt(3)/2 xx 12`

∴ `square` = 6 and BC = `6sqrt(3)`


If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.


Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reasons for your answer.


In ∆PQR, PD ⊥ QR such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b)(a – b) = (c + d)(c – d).


In the adjoining figure, a tangent is drawn to a circle of radius 4 cm and centre C, at the point S. Find the length of the tangent ST, if CT = 10 cm.


In an equilateral triangle PQR, prove that PS2 = 3(QS)2.


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 angles are said to be ______, if they have equal measures.


Two rectangles are congruent, if they have same ______ and ______.


In a triangle, sum of squares of two sides is equal to the square of the third side.


Two squares having same perimeter are congruent.


Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?


Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.


Share
Notifications



      Forgot password?
Use app×