# What is the orthocenter of a triangle with corners at #(4 ,7 )#, #(9 ,5 )#, and #(5 ,6 )#?

The orthocenter is the point where the extended altitudes of a triangle meet. This will be inside the triangle if the triangle is acute, outside of the triangle if the triangle is obtuse. In the case of the right angled triangle it will be at the vertex of the right angle. ( The two sides are each altitudes ).

It is generally easier is you do a rough sketch of the points so you know where you are.

Let

Since the altitudes pass through a vertex and are perpendicular to the side opposite, we need the find the equations of these lines. It will be obvious from the definition that we only need to find two of these lines. These will define a unique point. It is unimportant which ones you choose.

I will use:

Line

Line

For

First find the gradient of this line segment:

A line perpendicular to this will have a gradient that is the negative reciprocal of this:

This passes through

For

Passing through

The intersection of

Solving simultaneously:

Substituting in

Orthocenter:

Notice the the orthocenter is outside the triangle because it is obtuse. The altitude lines passing through

By signing up, you agree to our Terms of Service and Privacy Policy

To find the orthocenter of a triangle, we first need to find the altitude of each side of the triangle. The altitude is a perpendicular line from one vertex to the opposite side.

- Calculate the slopes of the lines containing each side of the triangle.
- Determine the perpendicular slopes to each side.
- Find the equations of the lines perpendicular to each side, passing through the opposite vertex.
- Find the intersection point of these perpendicular lines. This point is the orthocenter.

Let's denote the vertices of the triangle as A(4, 7), B(9, 5), and C(5, 6).

The slope of line AB: m_AB = (5 - 7) / (9 - 4) = -2/5. The slope of line BC: m_BC = (6 - 5) / (5 - 9) = 1/(-4) = -1/4. The slope of line AC: m_AC = (6 - 7) / (5 - 4) = -1.

The perpendicular slopes are the negative reciprocals of the original slopes: Perpendicular to AB: m_perp_AB = -1/m_AB = 5/2. Perpendicular to BC: m_perp_BC = -1/m_BC = 4. Perpendicular to AC: m_perp_AC = -1/m_AC = 1.

Now, we use point-slope form to find the equations of the lines passing through each vertex and perpendicular to the opposite side: Line perpendicular to AB passing through C: y - 6 = 1(x - 5) --> y = x + 1. Line perpendicular to BC passing through A: y - 7 = 5/2(x - 4) --> y = (5/2)x - 3. Line perpendicular to AC passing through B: y - 5 = 4(x - 9) --> y = 4x - 31.

Now, we solve the system of equations formed by these three lines to find the intersection point, which is the orthocenter of the triangle. This point is (5, 4). Therefore, the orthocenter of the triangle with vertices (4, 7), (9, 5), and (5, 6) is at coordinates (5, 4).

By signing up, you agree to our Terms of Service and Privacy Policy

The orthocenter of the triangle with corners at (4, 7), (9, 5), and (5, 6) is at the point (6.2, 5.8).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A line segment is bisected by a line with the equation # 3 y - 7 x = 2 #. If one end of the line segment is at #(7 ,8 )#, where is the other end?
- What is the orthocenter of a triangle with corners at #(4 ,7 )#, #(8 ,2 )#, and (5 ,6 )#?
- A triangle has corners A, B, and C located at #(4 ,7 )#, #(3 ,2 )#, and #(2 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- What is the orthocenter of a triangle with corners at #(4 ,5 )#, #(3 ,7 )#, and (5 ,6 )#?
- Let #P(a,b) and Q(c,d)# be two points in the plane. Find the equation of the line #l# that is the perpendicular bisector of the line segment #bar(PQ)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7