Kelas 8 - Perbandingan Sisi-Sisi Segitiga Siku-siku Istimewa

Special Right Triangles

The lesson explains how to use side-length ratios for two special right triangles and applies them to five example problems.

45°–45°–90° Triangle

Because the two acute angles are equal, the sides opposite them are equal. The triangle is therefore an isosceles right triangle.

The side ratio is:

[ \text{side opposite }45^\circ : \text{side opposite }45^\circ : \text{hypotenuse} = 1:1:\sqrt{2} ]

If each shorter side has length (x), the hypotenuse is (x\sqrt{2}).

30°–60°–90° Triangle

The side opposite the smallest angle, (30^\circ), is the shortest. The side opposite (90^\circ) is the hypotenuse.

The side ratio is:

[ \text{side opposite }30^\circ : \text{side opposite }60^\circ : \text{hypotenuse} = 1:\sqrt{3}:2 ]

If the shortest side is (x), the other sides are (x\sqrt{3}) and (2x).

Worked Examples

1. 45°–45°–90° triangle: If a shorter side is (\sqrt{3}), the hypotenuse is [ \sqrt{3}\times\sqrt{2}=\sqrt{6}. ]

2. 45°–45°–90° triangle: If the hypotenuse is (8\sqrt{2}), each equal leg is [ 8\sqrt{2}\div\sqrt{2}=8\text{ cm}. ]

3. 30°–60°–90° triangle: If the side opposite (30^\circ) is (3\sqrt{3}), the side opposite (60^\circ) is [ 3\sqrt{3}\times\sqrt{3}=9\text{ cm}, ] and the hypotenuse is [ 2(3\sqrt{3})=6\sqrt{3}\text{ cm}. ]

4. 45°–45°–90° triangle: If the hypotenuse is (20\text{ cm}), each leg is [ 20\div\sqrt{2}=10\sqrt{2}\text{ cm}. ]

5. 30°–60°–90° triangle: If the hypotenuse is (12\sqrt{3}), the side opposite (30^\circ) is half of it, or (6\sqrt{3}). The side opposite (60^\circ) is [ 6\sqrt{3}\times\sqrt{3}=18\text{ cm}. ]

Working with Square Roots

The lesson also reviews multiplying square roots, such as (\sqrt{3}\times\sqrt{3}=3), and simplifying division by a square root. For example:

[ \frac{5}{\sqrt{3}}=\frac{5\sqrt{3}}{3}. ]

Speakers or Sources

• One unnamed instructor or narrator presents the lesson and works through diagrams and examples.
• No other speakers are featured.