A triangle ABC with vertices A(-1,0), B(-2,3/4), C(-3,7/6) has its orthocentre H.Then the orthocentre of triangle BCH will be.?

To find the orthocenter of triangle BCH, we first need to determine the orthocenter H of triangle ABC. The orthocenter is the point where the three altitudes of the triangle intersect. Let's go through the steps to find H and then use it to find the orthocenter of triangle BCH.

Step 1: Find the Orthocenter H of Triangle ABC

Given the vertices of triangle ABC: A(-1, 0), B(-2, 3/4), and C(-3, 7/6), we will first calculate the slopes of the sides and then find the equations of the altitudes.

Calculating the Slope of Each Side

The slope of line segment AB can be calculated using the formula:

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

• Slope of AB: m_AB = (3/4 - 0) / (-2 + 1) = (3/4) / (-1) = -3/4
• Slope of BC: m_BC = (7/6 - 3/4) / (-3 + 2) = (7/6 - 9/12) / (-1) = (14/12 - 9/12) / (-1) = 5/12
• Slope of AC: m_AC = (7/6 - 0) / (-3 + 1) = (7/6) / (-2) = -7/12

Finding the Altitudes

Now, we need to find the equations of the altitudes from each vertex:

• Altitude from A is perpendicular to BC. The slope of altitude from A is the negative reciprocal of m_BC, which is -12/5. Using point-slope form: y - 0 = (-12/5)(x + 1) gives us the equation of altitude from A.
• Altitude from B is perpendicular to AC. The slope of this altitude is the negative reciprocal of m_AC, which is 12/7. Using point-slope form: y - (3/4) = (12/7)(x + 2).
• Altitude from C is perpendicular to AB. The slope of this altitude is the negative reciprocal of m_AB, which is 4/3. The equation is: y - (7/6) = (4/3)(x + 3).

Step 2: Solving the System of Equations

Next, we solve these equations simultaneously to find the coordinates of H. Let’s start by simplifying and solving the equations one step at a time. For instance, solving the first two altitude equations will give us the coordinates of H.

Step 3: Finding the Orthocenter of Triangle BCH

Once we have the coordinates for H, we can treat triangle BCH, where B(-2, 3/4), C(-3, 7/6), and H are the vertices. The same process as before will apply here to find the orthocenter of triangle BCH. You will need to calculate the slopes of sides BC and BH, and then find the equations of the altitudes from B and C. The orthocenter will be where these altitudes intersect.

Final Note on Orthocenters

Remember that the orthocenter can lie inside the triangle, on the triangle, or outside the triangle, depending on the type of triangle you are dealing with (acute, right, or obtuse). This characteristic can also help you visualize the position of H relative to triangles ABC and BCH.

By following these steps, you should be able to find the orthocenter of triangle BCH! If you need any specific calculations or further clarifications, feel free to ask!