Answer:
altitude ≈ 3.392
Explanation:
To find the length of the altitude of the triangle, we need to first determine which side of the triangle is perpendicular to the base. Let's call the vertices of the triangle A (0, 6), B (8, 0), and C (5, 8), and let's assume that the altitude is drawn from vertex C and is perpendicular to side AB.
The slope of the line AB is:
mAB = (y2 - y1)/(x2 - x1) = (0 - 6)/(8 - 0) = -3/4
The slope of a line perpendicular to AB is the negative reciprocal of its slope, which is:
m_perp = -1/mAB = -1/(-3/4) = 4/3
So, the equation of the line passing through point C with slope 4/3 is:
y - y1 = m_perp(x - x1)
y - 8 = (4/3)(x - 5)
y - 8 = (4/3)x - (20/3)
y = (4/3)x - (20/3) + 8
y = (4/3)x + 4/3
Now we can find the x-coordinate of the point where this line intersects with AB. To do this, we solve the system of equations:
y = (4/3)x + 4/3
y = -3/4x + 6
(4/3)x + 4/3 = -3/4x + 6
(25/12)x = 14/3
x = (14/3) ÷ (25/12) = 1.536
So, the point of intersection is (1.536, 4.608).
The altitude of the triangle is the perpendicular distance between point C and line AB, which is the y-coordinate of point C minus the y-coordinate of the point of intersection:
altitude = yC - y-coordinate of point of intersection
altitude = 8 - 4.608
altitude ≈ 3.392
Therefore, the length of the altitude of the triangle is approximately 3.392 units.