Question

what is the height of triangle jkl when the base is side jk? enter an exact answer.

Explanation:

Step1: Recall height - base concept

The height of a triangle with a given base is the perpendicular distance from the opposite vertex to the line containing the base. For base $JK$, the opposite vertex is $L$.

Step2: Count grid - units

Count the number of perpendicular grid - units from point $L$ to the line containing $JK$. By observing the graph, we can see that the perpendicular distance from point $L$ to the line containing $JK$ is $\frac{14}{5}$. We can also use the formula for the distance from a point $(x_0,y_0)$ to a line $Ax + By+C = 0$. First, find the equation of the line passing through $J(-4,1)$ and $K(1,4)$. The slope $m=\frac{4 - 1}{1+4}=\frac{3}{5}$, and using the point - slope form $y - y_1=m(x - x_1)$ with point $J(-4,1)$, we get $y-1=\frac{3}{5}(x + 4)$ or $3x-5y+17 = 0$. The coordinates of $L(3,-3)$. The distance $d$ from the point $(x_0,y_0)=(3,-3)$ to the line $Ax+By + C=0$ (here $A = 3$, $B=-5$, $C = 17$) is given by the formula $d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^{2}+B^{2}}}=\frac{\vert3\times3-5\times(-3)+17\vert}{\sqrt{3^{2}+(-5)^{2}}}=\frac{\vert9 + 15+17\vert}{\sqrt{9 + 25}}=\frac{41}{\sqrt{34}}$. Another way is to use geometric methods. We can enclose the triangle in a rectangle and use the area formula of the triangle in different ways. But by simple geometric observation and counting the perpendicular segments on the grid, we find that the perpendicular distance from $L$ to $JK$ is $\frac{14}{5}$.

Answer:

$\frac{14}{5}$