Altitude of Triangle

Introduction to Altitude of a Triangle

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the extended base of the altitude. The altitude is the shortest distance from the vertex to its opposite side. The 3 altitudes always meet at a single point, no matter what the shape of the triangle is. The point where the 3 altitudes meet is called the ortho-centre of the triangle. The altitude of a triangle is also known as the height of the triangle. In triangle ABC, AD is the altitude which is a perpendicular line drawn from the vertex A to the point D in the opposite side BC.

 Orthocenter in Altitude of a Triangle

The orthocenter is the point where all the three altitudes of the triangle cut or intersect each other. Here, the altitude is the line drawn from the vertex of the triangle and is perpendicular to the opposite side. Orthocenter indicates the center of all the right angles from the vertices to the opposite sides i.e., the altitudes. The term ortho means right and it is considered to be the intersection point of three altitudes drawn from the three vertices of a triangle. There is no direct formula to calculate the orthocenter of the triangle. It lies inside for an acute and outside for an obtuse triangle. Altitudes are nothing but the perpendicular line ( AD, BE and CF ) from one side of the triangle ( either AB or BC or CA ) to the opposite vertex

Properties of Altitude of a Triangle

There are a maximum of three altitudes for a triangle. The altitude of a triangle is perpendicular to the opposite side. Thus, it forms 90 degrees angle with the opposite side. Depending on the type of triangle, the altitude can lie inside or outside the triangle. The altitude of a triangle is the perpendicular line segment drawn from the vertex of the triangle to the side opposite to it. The altitude makes a right angle with the base of the triangle that it touches. The altitude is the shortest distance from the vertex to its opposite side. The 3 altitudes always meet at a single point, no matter what the shape of the triangle is. The point where the 3 altitudes meet is called the ortho-centre of the triangle. The basic formula to find the area of a triangle is: Area = 1/2 × base × height, where the height represents the altitude. Using this formula, we can derive the formula to calculate the height (altitude) of a triangle: Altitude = (2 × Area)/base.

Relation with Circles and Conics

The circle is the simplest and best known conic section. As a conic section, the circle is the intersection of a plane perpendicular to the cone's axis. is the circle's center also spelled as centre. The diameter (D) is twice the length of the radius. A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. The equation of a circle is (x - h)² + (y - k)² = r² where r is equal to the radius, and the coordinates (x,y) are equal to the circle center. The variables h and k represent horizontal or vertical shifts in the circle graph.

A conic section can also be described as the locus of a point P moving in the plane of a fixed point F known as focus (F) and a fixed line d known as directrix (with the focus not on d) in such a way that the ratio of the distance of point P from focus F to its distance from d is a constant e known as eccentricity. Now,

• If eccentricity, e = 0, the conic is a circle
• If 0<e<1, the conic is an ellipse
• If e=1, the conic is a parabola
• And if e>1, it is a hyperbola

So, eccentricity is a measure of the deviation of the ellipse from being circular. Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, the eccentricity is;

e = cos α/cos 

Relation to Other Centers, The Nine-Point Circle

The nine-point center forms the center of a point reflection that maps the medial triangle to the Euler triangle, and vice versa. According to Lester's theorem, the nine-point center lies on a common circle with three other points: the two Fermat points and the circumcenter. One way to find the center of the nine point circle (O) is to find the midpoint f the segment that connects the orthocenter (H) and the circumcenter (N), which is the intersection point of the perpendicular bisectors of each side of the original triangle.

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Most Important Frequently Asked Questions Searched By Students

Q: What is the Altitude of a Triangle in Math?

Answer: In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the extended base of the altitude.

Q: What is the Use of Altitude of a Triangle?

Answer: The altitude makes a right angle with the base of the triangle that it touches. It is commonly referred to as the height of a triangle and is denoted by the letter 'h'. It can be measured by calculating the distance between the vertex and its opposite side.

Q: What is the Difference Between the Median and Altitude of a Triangle?

Answer: A median of a triangle is a line segment that joins a vertex to the mid-point of the side that is opposite to that vertex. In the figure, AD is the median that divides BC into two equal halves, that is, DB = DC.

• Every triangle has 3 medians, one from each vertex. AE, BF, and CD are the 3 medians of the triangle ABC.
• The 3 medians always meet at a single point, no matter what the shape of the triangle is.
• The point where the 3 medians meet is called the centroid of the triangle. Point O is the centroid of the triangle ABC.
• Each median of a triangle divides the triangle into two smaller triangles that have equal area.
• In fact, the 3 medians divide the triangle into 6 smaller triangles of equal area.