Orthocentre of the triangle whose vertices are given by the coordinates(0,0),(3,4),(4,0)is

To find the orthocenter of a triangle given its vertices, we first need to understand what the orthocenter is. The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment from a vertex to the opposite side, forming a right angle with that side. For the triangle with vertices at (0,0), (3,4), and (4,0), we can calculate the orthocenter using the coordinates provided.

Step 1: Identify the Vertices

The vertices of the triangle are:

• A(0, 0)
• B(3, 4)
• C(4, 0)

Step 2: Find the Slopes of the Sides

To find the equations of the altitudes, we first need the slopes of the sides of the triangle. The slope is determined using the formula:

Slope (m) = (y2 - y1) / (x2 - x1)

Calculating the slopes:

• Slope of AB (from A to B): m_AB = (4 - 0) / (3 - 0) = 4/3
• Slope of BC (from B to C): m_BC = (0 - 4) / (4 - 3) = -4
• Slope of AC (from A to C): m_AC = (0 - 0) / (4 - 0) = 0

Step 3: Find the Slopes of the Altitudes

The slopes of the altitudes are the negative reciprocals of the slopes of the sides they are perpendicular to:

• Slope of altitude from C (perpendicular to AB): m_C = -3/4
• Slope of altitude from A (perpendicular to BC): m_A = 1/4
• Slope of altitude from B (perpendicular to AC): m_B = undefined (vertical line)

Step 4: Find the Equations of the Altitudes

Now we can write the equations of the altitudes using the point-slope form of a line, which is:

y - y1 = m(x - x1)

For the altitude from C(4,0):

y - 0 = -3/4(x - 4) → y = -3/4x + 3

For the altitude from A(0,0):

y - 0 = 1/4(x - 0) → y = 1/4x

For the altitude from B(3,4), since the slope is undefined, the equation is:

x = 3

Step 5: Find the Intersection of the Altitudes

To find the orthocenter, we need to solve the equations of two altitudes. We'll use the altitudes from C and A:

From C: y = -3/4x + 3

From A: y = 1/4x

Setting them equal to each other:

-3/4x + 3 = 1/4x

Combining like terms:

3 = 1x

x = 3

Now substituting x = 3 into one of the altitude equations (using y = 1/4x):

y = 1/4(3) = 3/4

The Orthocenter Coordinates

Thus, the coordinates of the orthocenter are (3, 3/4).

Final Thoughts

In summary, the orthocenter of the triangle with vertices at (0,0), (3,4), and (4,0) is located at the point (3, 3/4). This point represents the intersection of the three altitudes drawn from each vertex of the triangle. Understanding how to derive the orthocenter through these steps not only solidifies your grasp of triangle properties but also reinforces your skills in coordinate geometry.