Three vertices of a triangle are A(4, 3),B(1, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The centroid,orthocentre,incentre,and circumcentre of a triangle are collinear if the triangle is isosceles.For the triangle to be isosceles,the s

Vedclass · Accepted Answer

$B$ or $C$ both

Vedclass · Answer

(D) The centroid,orthocentre,incentre,and circumcentre of a triangle are collinear if the triangle is isosceles.For the triangle to be isosceles,the sides must satisfy $AB = BC$,$AB = AC$,or $BC = AC$.Case $1$: $AB = BC$$AB^2 = (4-1)^2 + (3-(-1))^2 = 3^2 + 4^2 = 9 + 16 = 25 \implies AB = 5$.$BC^2 = (7-1)^2 + (k-(-1))^2 = 6^2 + (k+1)^2 = 36 + (k+1)^2$.$36 + (k+1)^2 = 25 \implies (k+1)^2 = -11$ (No real solution).Case $2$: $AB = AC$$AC^2 = (7-4)^2 + (k-3)^2 = 3^2 + (k-3)^2 = 9 + (k-3)^2$.$9 + (k-3)^2 = 25 \implies (k-3)^2 = 16 \implies k-3 = \pm 4$.$k = 7$ or $k = -1$.Case $3$: $BC = AC$$36 + (k+1)^2 = 9 + (k-3)^2$.$36 + k^2 + 2k + 1 = 9 + k^2 - 6k + 9$.$37 + 2k = 18 - 6k \implies 8k = -19 \implies k = -19/8$.Since the triangle is isosceles for $k = 7, -1, -19/8$,the points are collinear in all these cases.

Three vertices of a triangle are $A(4, 3)$,$B(1, -1)$,and $C(7, k)$. The value$(s)$ of $k$ for which the centroid,orthocentre,incentre,and circumcentre of the $\Delta ABC$ lie on the same straight line is/are:

Similar Questions

In a triangle $ABC$,$\angle BAC = 90^{\circ}$; $AD$ is the altitude from $A$ onto $BC$. Draw $DE$ perpendicular to $AC$ and $DF$ perpendicular to $AB$. Suppose $AB = 15$ and $BC = 25$. Then the length of $EF$ is

If a straight line $L$ is perpendicular to the line $4x - 2y = 1$ and forms a triangle of area $4 \text{ sq unit}$ with the coordinate axes,then the equation of the line $L$ is

If the extremities of the base of an isosceles triangle are the points $(2a, 0)$ and $(0, a)$ and the equation of one of the sides is $x = 2a$,then the area of the triangle is

$\Delta ABC$ has integral side lengths. The internal angle bisector of angle $B$ meets side $AC$ at $D$. If $AD = 3$ and $DC = 8$,then the perimeter of $\Delta ABC$ is:

The bisectors of the angles of a parallelogram enclose a

Vedclass Test Series

Exam Paper Generator

Online Exam Module