PLEASE HELP, find the orthocenter of this triangle

The orthocenter of a triangle is the point at which the three altitudes of a triangle intersect.  This involves creating 3 lines, one from each vertex (K, L, and M) perpendicular to the line segment opposite it, so K⊥LM, L⊥KM, and M⊥KL.

Because these three lines intersect at one point, you only need to do this procedure for two of the three altitudes.  Because K and M are on the same Y coordinate (-4), lets do this for LM and KL

First, we need to find the slope of the perpendicular to each line segment, so we need the slope of LM and KL.

The slope is the change in Y over the change in X

Slope LM = 6 -(-4) / 2-7 = 10/-5 = -2

Slope KL =  -4-(6) / 4-2 = -10/2 = -5

The slope of a perpendicular line is the negative reciprocal of the slope of the given line, so for

the slope of K⊥LM has to be 1/2 and

the slope of M⊥KL has to be 1/5

Second, the vertex has to be on a line with that slope.  So using the point-slope formula

(y-y1) = m(x-x1) , where x1,y1 is a point on the line for each line gives us

K⊥LM:  y-(-4) = 1/2(x-4)

M⊥KL:  y-(-4) = 1/5(x-7)

Third, find the point on both lines, are where the two equations are equal

So, 1/2(x)-2 = 1/5(x) - 7/5;

5x - 20 = 2x - 14

3x = 6 --> x=2

y+4=-1 --> y=-5

(2,-5)

Step-by-step explanation:

there are many solutions.

step-by-step explanation:

step 1: simplify both sides of the equation.

4x+6=4x^2+10x−4

step 2: subtract 4x^2+10x-4 from both sides.

4x+6−(4x^2+10x−4)=4x^2+10x−4−(4x^2+10x−4)

−4x^2−6x+10=0

step 3: factor left side of equation.

−2(2x+5)(x−1)=0

step 4: set factors equal to 0.

x=-5/2 or x=1