45-45-90 triangle side ratios | Right triangles and trigonometry | Geometry | Khan Academy

TL;DR

The video explains the ratios of the sides in 30-60-90 and 45-45-90 triangles, which can be used to determine the lengths of the sides.

Transcript

In the last video, we showed that the ratios of the sides of a 30-60-90 triangle are-- if we assume the longest side is x, if the hypotenuse is x. Then the shortest side is x/2 and the side in between, the side that's opposite the 60 degree side, is square root of 3x/2. Or another way to think about it is if the shortest side is 1-- Now I'll do the... Read More

Key Insights

• 🙃 The ratios of the sides in a 30-60-90 triangle are x, x/2, and √3x/2.
• ☺️ The ratios of the sides in a 45-45-90 triangle are x, x, and √2x.
• 🥳 These ratios allow for easy determination of side lengths in similar triangles.
• 🥳 A 30-60-90 triangle can be identified by the ratios 1, √3, and 2.
• 🦿 A 45-45-90 triangle is an isosceles right triangle, with equal legs and a hypotenuse equal to √2 times the leg length.
• 🥳 The Pythagorean theorem can be used to derive the ratios in both types of triangles.
• 🥳 Understanding these ratios is useful in geometry and trigonometry.

Questions & Answers

Q: What are the ratios of the sides in a 30-60-90 triangle?

In a 30-60-90 triangle, the ratios of the sides are x, x/2, and √3x/2. The longest side (hypotenuse) is x, the shortest side is x/2, and the side opposite the 60-degree angle is √3x/2.

Q: How do you determine if a triangle is a 30-60-90 triangle?

If a triangle has the ratios of sides 1, √3, and 2, it is a 30-60-90 triangle. Alternatively, if you know the side opposite the 30-degree angle is 1, then the side opposite the 60-degree angle is √3, and the hypotenuse is 2, it is a 30-60-90 triangle.

Q: What is the ratio of sides in a 45-45-90 triangle?

In a 45-45-90 triangle, the ratios of the sides are x, x, and √2x. Both legs have the same length, and the hypotenuse is √2 times the length of either of the legs.

Q: How can you determine the lengths of the sides in a 45-45-90 triangle?

If you know the length of one leg in a 45-45-90 triangle, you can find the lengths of the other leg and the hypotenuse by using the ratios 1, 1, and √2.

Summary & Key Takeaways

• The ratios of the sides in a 30-60-90 triangle are x, x/2, and √3x/2.

• The ratios of the sides in a 45-45-90 triangle are x, x, and √2x.

• These ratios can be used to find the lengths of the sides in similar triangles.