Special Right Triangles

Two triangles worth knowing by heart.

12 problems · about 20 minutes · G-SRT.C.8

What this lesson teaches

The student uses the 45-45-90 and 30-60-90 triangles to find exact trigonometric values.

The 45-45-90 triangle comes from a square; its hypotenuse is leg × √2.
The 30-60-90 triangle comes from an equilateral triangle; the short leg is half the hypotenuse.
These two triangles supply every exact value you are expected to know.

Straightforward practice. Get the method working first.

5 problems

In a 30-60-90 triangle, the hypotenuse is 20. What is the short leg?
Answer 10
Why 10.
Which triangle comes from cutting a square in half?
Answer The 45-45-90 triangle.
Why The 45-45-90 triangle.
sin 30° as a decimal?
Answer 0.5
Why Half the hypotenuse.
cos 60° as a decimal?
Answer 0.5
Why Cosine of 60° matches sine of 30°.
tan 45° as a decimal?
Answer 1
Why Equal legs.

The same ideas with more to keep track of.

3 problems

A 30-60-90 triangle has short leg 5. What is the hypotenuse?
Answer 10
Why Double the short leg.
A 45-45-90 triangle has legs 7. Hypotenuse to two decimal places?
Answer 9.9
Why 7 × 1.414.
sin 60° to three decimal places?
Answer 0.866
Why √3 ÷ 2.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each angle with its sine.
Answer 30° → 1/2; 45° → √2/2; 60° → √3/2
Why The sines climb as the angle grows.
A 30-60-90 triangle has short leg 4. Long leg to two decimal places?
Answer 6.93
Why 4 × √3.
The Square: A square has sides of 10. What is the diagonal, to two decimal places?
Answer 14.14
Why 10√2, about 14.14.
The Ramp: A 30-60-90 triangle has hypotenuse 12. What is the short leg?
Answer 6
Why 6.