Area of Oblique Triangles ENRICHMENT

Front

What is the formula for the area of a triangle when two sides and the included angle are known?

Back

The area of a triangle can be calculated using 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 Heron's formula for the area of a triangle?

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

How do you find the area of an oblique triangle given all three sides?

Back

To find the area of an oblique triangle with sides \( a \), \( b \), and \( c \), use Heron's formula: first calculate the semi-perimeter \( s \), then apply the formula \( A = \sqrt{s(s-a)(s-b)(s-c)} \).

Front

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

Back

The sine function is used in the area formula \( A = \frac{1}{2}ab \sin(C) \) to account for the angle between the two sides, which affects the height of the triangle.

Front

If a triangle has sides of lengths 7, 8, and 9, what is its area using Heron's formula?

Back

First, calculate the semi-perimeter: \( s = \frac{7+8+9}{2} = 12 \). Then, apply Heron's formula: \( A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} \approx 26.83 \).

Front

What is the area of a triangle with a base of 10 units and a height of 5 units?

Back

The area can be calculated using the formula: \( A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 5 = 25 \) square units.

Front

How do you find the area of a triangle when given two sides and the angle between them?

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 angle between them.

Front

What is the area of Triangle ABC if a = 29 in, b = 25 in, c = 27 in?

Back

Using Heron's formula: \( s = \frac{29+25+27}{2} = 40.5 \). Then, \( A = \sqrt{40.5(40.5-29)(40.5-25)(40.5-27)} = \sqrt{40.5 \cdot 11.5 \cdot 15.5 \cdot 13.5} \approx 312.2 \) square inches.

Front

What is the relationship between the area of a triangle and its height?

Back

The area of a triangle is directly proportional to its height. As the height increases, the area increases, given a constant base.

Front

What is the formula for the area of a triangle using coordinates of its vertices?

Back

The area can be calculated using the formula: \( A = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \), where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices.