Acute Angle Triangle

Acute Angle Triangle Definition

Any triangle that has an acute angle as all of its interior angles is defined as an acute angle triangle. (An acute angle is an angle less than 90°

Acute Angle Example

Consider \(\Delta ABC\) in the figure below. The angles formed by the intersection of lines AB, BC and CA are \(\angle ABC\), \(\angle BCA\), and \(\angle CAB\) respectively. We can see that

\(\angle ABC\) = \(75^{\circ}\) \(\angle BCA\) = \(65^{\circ}\) \(\angle CAB\) = \(40^{\circ}\)

Since all the three angles are less than 90°, we can infer that \(\Delta ABC\) is an acute angle triangle or acute angled triangle.

Circumcenter

A perpendicular bisector is a segment that divides any side of a triangle into two equal parts. The intersection of perpendicular bisectors of all the three sides of an acute angled triangle forms the circumcenter and it always lies inside the triangle.

Incenter

An angular bisector is a segment that divides any angle of a triangle into two equal parts. The intersection of angular bisectors of all the three angles of an acute angle triangle forms the incenter and it always lies inside the triangle.

Centroid

A median of a triangle is the line that connects an apex with the midpoint of the opposite side. In acute angle triangle, the medians intersect at the centroid of the triangle and it always lies inside the triangle.

Orthocenter

An altitude of a triangle is a line that passes through an apex of a triangle and is perpendicular to the opposite side. The three altitudes of an acute angle triangle intersect at the orthocenter and it always lies inside the triangle.

Distance Between Orthocenter and Circumcenter

For an acute angle triangle, the distance between orthocenter and circumcenter is always less than the circumradius.

Square of the Longest Side

For an acute angle triangle, the square of the longest side will always be less than the sum of the squares of the other two sides.

Construction of Acute Angle Triangle

The triangle can be constructed if the following are given:

  • Length of all three sides
  • Angular measurement of all three angles
  • Length of any two sides and the angular measurement of the included angle
  • Length of any one side and angular measurement of two base angles

Acute Angled Triangle Practice Questions

  1. If two angles of an acute angled triangle are \(85^{\circ}\) and \(30^{\circ}\), what is the angular measurement of the third angle?
  2. Find the area of the triangle if the length of one side is 8 cms and the corresponding altitude is 6 cms.
  3. Construct an acute angle triangle which has a base of 7 cms and base angles 65^{\circ} and 75^{\circ}. Find the circumcenter and orthocenter.

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Practise This Question

 Let R be a relation on the set N be defined by {(x,y)|x,y|N,2x+y=41}. Then R is