Area of Oblique Triangles

Front

What is the area formula for an oblique triangle?

Back

The area of an oblique triangle can be calculated using the formula: \( A = \frac{1}{2}ab \sin(C) \), where \( a \) and \( b \) are the lengths of two sides and \( C \) is the included angle.

Front

What defines an oblique triangle?

Back

An oblique triangle is defined as a triangle that does not contain a right angle (90°). It can either be acute (all angles less than 90°) or obtuse (one angle greater than 90°).

Front

How do you find the area of a triangle using Heron's formula?

Back

Heron's formula states that the area \( A \) of a triangle with sides of lengths \( a \), \( b \), and \( c \) is given by: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \) is the semi-perimeter.

Front

What is the semi-perimeter of a triangle?

Back

The semi-perimeter \( s \) of a triangle is half the sum of the lengths of its sides: \( s = \frac{a+b+c}{2} \).

Front

If a triangle has sides of lengths 7, 8, and 9, what is its semi-perimeter?

Back

The semi-perimeter \( s \) is \( \frac{7+8+9}{2} = 12 \).

Front

What is the significance of the sine function in calculating the area of a triangle?

Back

The sine function relates the angle between two sides of a triangle to the area, allowing for the calculation of the area when two sides and the included angle are known.

Front

How do you calculate the area of an oblique triangle given two sides and the included angle?

Back

Use the formula: \( A = \frac{1}{2}ab \sin(C) \), where \( a \) and \( b \) are the lengths of the two sides and \( C \) is the included angle.

Front

What is the area of a triangle with sides 10 cm, 12 cm, and an included angle of 30°?

Back

Using the formula: \( A = \frac{1}{2} \times 10 \times 12 \times \sin(30°) = 60 \text{ cm}^2 \).

Front

What is the relationship between the angles and sides in an oblique triangle?

Back

In an oblique triangle, the larger the angle, the longer the opposite side. This is known as the Law of Sines.

Front

What is the Law of Sines?

Back

The Law of Sines states that \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \), where \( a, b, c \) are the lengths of the sides opposite angles \( A, B, C \) respectively.