In the triangle ABC with vertices A(2, 3),B(4 | vedclass

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

Question

(A) Let $AD$ be the altitude of triangle $ABC$ from vertex $A$. Accordingly,$AD \perp BC$. The slope of $BC$ is $m_{BC} = \frac{2 - (-1)}{1 - 4} = \fr

Vedclass · Accepted Answer

$y-x=1$ and $\sqrt{2}$

Vedclass · Answer

(A) Let $AD$ be the altitude of triangle $ABC$ from vertex $A$. Accordingly,$AD \perp BC$. The slope of $BC$ is $m_{BC} = \frac{2 - (-1)}{1 - 4} = \frac{3}{-3} = -1$. Since $AD \perp BC$,the slope of $AD$ is $m_{AD} = -\frac{1}{m_{BC}} = -\frac{1}{-1} = 1$. The equation of the line $AD$ passing through point $A(2, 3)$ with slope $1$ is: $(y - 3) = 1(x - 2)$ $\Rightarrow y - 3 = x - 2$ $\Rightarrow y - x = 1$. Length of $AD$ is the perpendicular distance from $A(2, 3)$ to the line $BC$. The equation of $BC$ is $(y - (-1)) = -1(x - 4)$ $\Rightarrow y + 1 = -x + 4$ $\Rightarrow x + y - 3 = 0$. The perpendicular distance $d$ from $(x_1, y_1)$ to $Ax + By + C = 0$ is $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. For $A(2, 3)$ and $x + y - 3 = 0$: $d = \frac{|1(2) + 1(3) - 3|}{\sqrt{1^2 + 1^2}} = \frac{|2 + 3 - 3|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$. Thus,the equation is $y - x = 1$ and the length is $\sqrt{2}$ units.

In the triangle $ABC$ with vertices $A(2, 3)$,$B(4, -1)$ and $C(1, 2)$,find the equation and length of the altitude from the vertex $A$.

Similar Questions

The orthocentre of the triangle formed by the lines $xy = 0$ and $x + y = 1$ is

If the coordinates of the orthocenter and the centroid of a triangle are $(4, -1)$ and $(2, 1)$ respectively,then what are the coordinates of the circumcenter of the triangle?

The medians $AD$ and $BE$ of a triangle with vertices $A(0, b)$,$B(0, 0)$,and $C(a, 0)$ are perpendicular to each other,if

If the coordinates of the vertices of the triangle $ABC$ are $A(-1, 6)$,$B(-3, -9)$,and $C(5, -8)$,then the equation of the median through $C$ is

If $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$ are the vertices of a triangle with side lengths $a, b, c$ opposite to vertices $A, B, C$ respectively,then the excentre with respect to vertex $B$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module