if the area is 12, the base is 8 and the height is 3, can the triangle be an acute triangle?

Question

How can you determine if a triangle is acute, obtuse or right if only height, base and area is known?

Answer 1

A triangle where all three internal angles are acute(less than 90 degrees ).

In any triangle, two of the interior angles are always acute (less than 90 degrees)^*, so there are three possibilities for the third angle:

• Less than 90° - all three angles are acute and so the triangle is acute.
• Exactly 90° - it is a right triangle
• Greater than 90° (obtuse): the triangle is an obtuse triangle

Area of the acute tringle is = 1 / 2 bh,

Area of the obtuse tringle is = 1 / 2 bh, and

Area of the right tringle is = 1 / 2 bh.

We can determine wheather the triangle is acute, obtuse, or right, only when the angle is given.

So, we can' t determine, wheather the triangle is acute, obtuse, or right, with base is 8, height is 3, and the area is 12.

Comments

In any triangle, two of the interior angles are always acute (less than 90 degrees)^*, so there are three possibilities for the third angle:

• Less than 90° - all three angles are acute and so the triangle is acute.
• Exactly 90° - it is a right triangle
• Greater than 90° (obtuse): the triangle is an obtuse triangle

The oblique triangle means a triangle that contains no right angle.

Therefore that triangle is either acute triangle or obtuse triangle.

The formula for the area of oblique triangle is Area = 1/2 bc sin A = 1/2 ab sin C = 1/2 ac sin B.

if angle A is 90^o, the formula gives the area for a right triangle :

Area = 1/2 bc sin A = 1/2 bc sin 90^o = 1/2 bc = 1/2 (base)(height).

The base of the triangle is 8 units and height is 3 units and its area is 12 square units.

To check  the above triangle is right angle triangle, as follows :

Area = 1/2 (base)(height).

12 = 1/2 (8)(3)

12 = 24/2

12 = 12.

The above statement is true, so the given triangle is right angle triangle.

To tell  the above triangle is right angle triangle, by use another method as follows :

Let the sides of the triangle are a = 8 units, b = 3 units and c.

Here to find the remaining side by use the Pythagorean Theorem, (hypotenuse)² = (opposite)² + (adjacent)²

c = √[ (3)² + (8)² ] = √[ 9 + 64 ] = √(73).

a = 8, b = 3 and c = √(73) ≅ 8.544.

a² = 64, b² = 9 and c² = 73.

The formula for the area of oblique triangle is Area = 1/2 bc sin A = 1/2 ab sin C = 1/2 ac sin B.

Find angle A:

Area = 1/2 bc sin A

12 = (1/2) (3)(8.544) sin A

24 = (3)(8.544)sin A

8/(8.544) = sin A

0.936 = sin A

A ≅ 69.444

Find angle B:

Area = 1/2 ac sin B

12 = (1/2) (8)(8.544) sin B

3 = (8.544)sin B

3/(8.544) = sin B

0.351 = sin B

B ≅ 20.556.

The remaining angle is C = 180^o - (69.444^o + 20.556^o) = 180^o - 90^o = 90^o.

The given triangle is right angle triangle.