Wednesday, 2 August 2017

Triangles VIII - Derivation of Acute Triangle and Obtuse Triangle Formulas

In this post, we will derive formulas to solve questions asked on an acute triangle and obtuse triangle for SSC Exams.
Acute triangle

Geometric Questions on Acute Triangle and Obtuse Triangle are frequently asked in SSC Exams. These questions can be easily solved if you know the formula of an acute triangle and obtuse triangle. In this post, we will derive the formula to make it easy for you to remember them while you are solving geometric questions on an acute triangle and obtuse triangle. 

In our previous post, we have discussed properties of right angled triangle, quickly revise those properties to have a better understanding of the formulas for the acute triangle and obtuse triangle.

Formula for Acute Triangle

An acute triangle is a triangle which has all the angles less than 90°.

Acute triangle

In right angled triangle the length of the largest side is measured by using the Pythagoras theorem.
Acute triangle

Acute triangle

In acute triangle since all the angles are less than 90°, the length of the largest side will be a little less than the right angled triangle.

Derivation:

In triangle ABC, when we draw a perpendicular to the side BC, we get two right angled-triangles, i.e., Triangle ADC and ADB.

Acute triangle

In triangle ADC 
(i) AC2 = AD2 + DC2 
(ii) DC2 = (BC – BD)2

By substituting (ii) in (i) we get,
AC2 = AD2 + (BC – BD)2

By using (a - b)2 = a2 + b2 - 2ab
(iii) AC2 = AD2 + BC2 + BD2 – 2(BC x BD)

According to Pythagoras theorem,
(iv) AD2 BD2 = AB2

By substituting (iv) in (iii) we get,
Acute triangle

Formula for Obtuse Triangle

An obtuse triangle is a triangle which has one angle more than 90°.

Acute triangle

In right angled triangle the length of the largest side is measured by using the Pythagoras theorem.
Acute triangle
Acute triangle

In an obtuse triangle, since one angle is more than 90°, the length of the largest side will be a little more than the right angled triangle.

Derivation:

In triangle ABC, when we draw a perpendicular to the extended side BC we get two right angled-triangles, i.e., Triangle ADC and ADB.

Acute triangle

In triangle ADC 
(i) AC2 = AD+ DC2 
(ii) DC2 = (BC + BD)2

By substituting (ii) in (i) we get,
AC2 = AD+ (BC + BD)2

By using (a + b)2 = a2 + b2 + 2ab
(iii) AC2 = AD+ BC+ BD+ 2(BC x BD)

According to Pythagoras theorem,
(iv) AD2 BD2 = AB2

By substituting (iv) in (iii) we get,

Acute triangle

Hope now it's easy for you to remember all the formulas for an Acute triangle and obtuse triangle. Do write in the comment section below on how these formulas have helped you solve geometric questions for SSC Exams. 

Practicing geometric questions will help you to remember these formulas of an acute triangle and obtuse triangle. To practice them, download free questions. 

Stay tuned for the next post.

Acute triangle
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