###### Advertisements

###### Advertisements

Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.

###### Advertisements

#### Solution

Let A (a, a), B(–a, –a) and C(– √3 a, √3 a) be the given points. Then, we have

`AB=\sqrt{(-a-a)^{2}+(-a-a)^{2})=sqrt(4a^2+4a^2) = 2sqrt2a`

`BC=sqrt((-sqrt3a+a)^2+(sqrt3a+a)^2)`

`=>BC=sqrt(a^2(1-sqrt3)^2+a^2(sqrt3+1)^2`

`\Rightarrow BC=a\sqrt{1+3-2\sqrt{3}+1+3+2\sqrt{3}}=a\sqrt{8}=2\sqrt{2}a and,\text{ }AC=\sqrt{(-\sqrt{3}a-a^2)+(\sqrt{3}a-a)}^{2}`

`=>AC=sqrt(a^2(sqrt3+1)+a^2(sqrt3-1)^2)`

`\Rightarrow AC=\sqrt{3+1+2\sqrt{3}+3+1-2\sqrt{3}}=a\sqrt{8}=2\sqrt{2}a`

Clearly, we have

AB = BC = AC

Hence, the triangle ABC formed by the given points is an equilateral triangle.

Now,

`Area of ∆ABC = \frac { \sqrt { 3 } }{ 4 } (side)^2`

`⇒ Area of ∆ABC = \frac { \sqrt { 3 } }{ 4 } × AB^2`

`⇒ Area of ∆ABC = \frac { \sqrt { 3 } }{ 4 } × (2√2 a)^2 sq. units = 2√3 a^2 sq. units`

#### RELATED QUESTIONS

Find the distance between the following pairs of points:

(−5, 7), (−1, 3)

The value of 'a' for which of the following points A(a, 3), B (2, 1) and C(5, a) a collinear. Hence find the equation of the line.

Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find

1) The ratio in which AB is divided by y-axis.

2) Find the coordinates of the point of intersection.

3) The length of AB.

Find the value of *a* when the distance between the points (3, *a*) and (4, 1) is `sqrt10`

Find the circumcenter of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6).

If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.

Find the co-ordinates of points of trisection of the line segment joining the point (6, -9) and the origin.

Find the distance of the following points from the origin:

(ii) B(-5,5)

`" Find the distance between the points" A ((-8)/5,2) and B (2/5,2)`

**Find the distance between the following pair of point.**

P(–5, 7), Q(–1, 3)

Find the distance between the following pair of points.

R(0, -3), S(0, `5/2`)

Find the distance between the following pair of points.

L(5, –8), M(–7, –3)

**Find the distance between the following pairs of point.**

W `((- 7)/2 , 4)`, X (11, 4)

Find the distances between the following point.

A(a, 0), B(0, a)

If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance

2AB is equal to

Distance of point (-3, 4) from the origin is .....

(A) 7 (B) 1 (C) 5 (D) 4

Find the distance between the following pairs of point in the coordinate plane :

(4 , 1) and (-4 , 5)

Find the distance between the following point :

(sin θ , cos θ) and (cos θ , - sin θ)

Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.

Prove that the points (4 , 6) , (- 1 , 5) , (- 2, 0) and (3 , 1) are the vertices of a rhombus.

ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.

ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.

From the given number line, find d(A, B):

Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.

Find the distance between the following pairs of points:

(-3, 6) and (2, -6)

Find the distance between the origin and the point:

(-8, 6)

Find the distance between the origin and the point:

(-5, -12)

Find the distance between the origin and the point:

(8, -15)

Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).

Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).

A point P lies on the x-axis and another point Q lies on the y-axis.

Write the ordinate of point P.

A point P lies on the x-axis and another point Q lies on the y-axis.

Write the abscissa of point Q.

Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.

Show that (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus.

Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.

The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.

Point P (2, -7) is the center of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of: AT

Point P (2, -7) is the centre of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of AB.

The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.

Find the distance of the following points from origin.

(5, 6)

Find distance between point A(– 3, 4) and origin O

Find distance between point Q(3, – 7) and point R(3, 3)

**Solution:** Suppose Q(x_{1}, y_{1}) and point R(x_{2}, y_{2})

x_{1} = 3, y_{1} = – 7 and x_{2} = 3, y_{2} = 3

Using distance formula,

d(Q, R) = `sqrt(square)`

∴ d(Q, R) = `sqrt(square - 100)`

∴ d(Q, R) = `sqrt(square)`

∴ d(Q, R) = `square`

Show that the points (2, 0), (– 2, 0) and (0, 2) are vertices of a triangle. State the type of triangle with reason

Show that A(1, 2), (1, 6), C(1 + 2 `sqrt(3)`, 4) are vertices of a equilateral triangle

Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.

∆ABC with vertices A(–2, 0), B(2, 0) and C(0, 2) is similar to ∆DEF with vertices D(–4, 0), E(4, 0) and F(0, 4).

Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).

What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?

If (a, b) is the mid-point of the line segment joining the points A(10, –6) and B(k, 4) and a – 2b = 18, find the value of k and the distance AB.

Find distance between points P(– 5, – 7) and Q(0, 3).

By distance formula,

PQ = `sqrt(square + (y_2 - y_1)^2`

= `sqrt(square + square)`

= `sqrt(square + square)`

= `sqrt(square + square)`

= `sqrt(125)`

= `5sqrt(5)`

What is the distance of the point (– 5, 4) from the origin?

Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.