If the midpoints of the sides of a triangle a | vedclass

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

Question

(C) Let the vertices of the triangle be $A(x_1, y_1), B(x_2, y_2),$ and $C(x_3, y_3)$.The midpoints are $M_1(0, 1), M_2(1, 1),$ and $M_3(1, 0)$.Using

Vedclass · Accepted Answer

$2 - \sqrt{2}$

Vedclass · Answer

(C) Let the vertices of the triangle be $A(x_1, y_1), B(x_2, y_2),$ and $C(x_3, y_3)$.The midpoints are $M_1(0, 1), M_2(1, 1),$ and $M_3(1, 0)$.Using the midpoint formula,we have:$\frac{x_1+x_2}{2} = 0, \frac{y_1+y_2}{2} = 1$$\frac{x_2+x_3}{2} = 1, \frac{y_2+y_3}{2} = 1$$\frac{x_3+x_1}{2} = 1, \frac{y_3+y_1}{2} = 0$Solving these,we get the vertices as $A(0, 0), B(0, 2),$ and $C(2, 0)$.The side lengths are $a = BC = \sqrt{(2-0)^2 + (0-2)^2} = \sqrt{4+4} = 2\sqrt{2}$,$b = AC = \sqrt{(2-0)^2 + (0-0)^2} = 2$,and $c = AB = \sqrt{(0-0)^2 + (2-0)^2} = 2$.The $x$-coordinate of the incenter $I$ is given by $x_I = \frac{ax_1 + bx_2 + cx_3}{a+b+c}$.$x_I = \frac{(2\sqrt{2})(0) + (2)(0) + (2)(2)}{2\sqrt{2} + 2 + 2} = \frac{4}{4 + 2\sqrt{2}} = \frac{2}{2 + \sqrt{2}}$.Rationalizing the denominator: $x_I = \frac{2(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} = \frac{2(2 - \sqrt{2})}{4 - 2} = \frac{2(2 - \sqrt{2})}{2} = 2 - \sqrt{2}$.

If the midpoints of the sides of a triangle are $(0, 1), (1, 1),$ and $(1, 0)$,what is the $x$-coordinate of the incenter of the triangle?

Similar Questions

Which one of the following statements is true?

If the sides $AB, BC, CD$ and $DA$ of a quadrilateral are given by the equations $x + 2y = 3, x = 1, x - 3y = 4$ and $5x + y + 12 = 0$ respectively,then find the angle between the diagonals $AC$ and $BD$ in degrees.

The sides of a quadrilateral are all positive integers and three of them are $5, 10, 20$. How many possible values are there for the fourth side?

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P, Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

Consider four triangles with side lengths $(5, 12, 9)$,$(5, 12, 11)$,$(5, 12, 13)$,and $(5, 12, 15)$. Among these,the triangle with the maximum area has sides:

Vedclass Test Series

Exam Paper Generator

Online Exam Module