Advertisements
Advertisements
Question
The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Then the other two sides are y – 3 = `(2 +- sqrt(3)) (x - 2)`.
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Let ABC be an equilateral triangle with vertex (2, 3) and the opposite side is x + y = 2 with slope –1.
Suppose slope of line AB is m.
Since each angle of equilateral triangle is 60°.
∴ Angle between AB and BC
tan 60° = `|(-1 - m)/(1 + (-1)m)|`
⇒ `sqrt(3) = |(1 + m)/(1 - m)|`
⇒ `sqrt(3) = +- ((1 + m)/(1 - m))`
Taking (+) sign,
`sqrt(3) = (1 + m)/(1 - m)`
⇒ `sqrt(3) - sqrt(3)m = 1 + m`
⇒ `sqrt(3)m + m = sqrt(3) - 1`
⇒ `m(sqrt(3) + 1) = sqrt(3) - 1`
⇒ `m = (sqrt(3) - 1)/(sqrt(3) + 1)
⇒ `m = (sqrt(3) - 1)/(sqrt(3) + 1) xx (sqrt(3) - 1)/(sqrt(3) - 1)`
⇒ `m = (3 + 1 - 2sqrt(3))/(3 - 1)`
= `2 - sqrt(3)`
Taking (–) sign,
`m = 2 + sqrt(3)`
So, the equations of other two lines are y – 3 = `(2 +- sqrt(3))(x - 2)`
APPEARS IN
RELATED QUESTIONS
The base of an equilateral triangle with side 2a lies along they y-axis such that the mid point of the base is at the origin. Find vertices of the triangle.
Find the distance between P (x1, y1) and Q (x2, y2) when :
- PQ is parallel to the y-axis,
- PQ is parallel to the x-axis
Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, –4) and B (8, 0).
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.
Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of a parallelogram.
Find the angle between the x-axis and the line joining the points (3, –1) and (4, –2).
If three point (h, 0), (a, b) and (0, k) lie on a line, show that `q/h + b/k = 1`
Consider the given population and year graph. Find the slope of the line AB and using it, find what will be the population in the year 2010?
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[- \frac{\pi}{4}\]
Find the slope of the lines which make the following angle with the positive direction of x-axis: \[\frac{\pi}{3}\]
What can be said regarding a line if its slope is negative?
Consider the following population and year graph:
Find the slope of the line AB and using it, find what will be the population in the year 2010.
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
Find the equations of the bisectors of the angles between the coordinate axes.
Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
Find the equations of the altitudes of a ∆ ABC whose vertices are A (1, 4), B (−3, 2) and C (−5, −3).
Find the angles between the following pair of straight lines:
3x + y + 12 = 0 and x + 2y − 1 = 0
Find the angles between the following pair of straight lines:
3x − y + 5 = 0 and x − 3y + 1 = 0
Find the angles between the following pair of straight lines:
3x + 4y − 7 = 0 and 4x − 3y + 5 = 0
Find the angles between the following pair of straight lines:
(m2 − mn) y = (mn + n2) x + n3 and (mn + m2) y = (mn − n2) x + m3.
Find the acute angle between the lines 2x − y + 3 = 0 and x + y + 2 = 0.
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.
The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is
The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is
If m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
If the slopes of the lines given by the equation ax2 + 2hxy + by2 = 0 are in the ratio 5 : 3, then the ratio h2 : ab = ______.
Point of the curve y2 = 3(x – 2) at which the normal is parallel to the line 2y + 4x + 5 = 0 is ______.
A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). Find the coordinates of the point A.
Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of 120° with the positive direction of x-axis.
A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2, 0), (0, 2) and (1, 1) on the line is zero. Find the coordinates of the point P.
Equation of the line passing through (1, 2) and parallel to the line y = 3x – 1 is ______.
Equations of diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y = 1 are ______.
One vertex of the equilateral triangle with centroid at the origin and one side as x + y – 2 = 0 is ______.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
The points A(– 2, 1), B(0, 5), C(– 1, 2) are collinear.
The line which passes through the origin and intersect the two lines `(x - 1)/2 = (y + 3)/4 = (z - 5)/3, (x - 4)/2 = (y + 3)/3 = (z - 14)/4`, is ______.
