Virginia SOL Math / Geometry / Triangles

G.TR.4

Right Triangles: Pythagorean Theorem & Trig

What you'll learn

  • Apply the Pythagorean Theorem to find a missing side
  • Identify the sine, cosine, and tangent ratios
  • Solve right triangles for a missing side or angle

Words to know

Hypotenuse —
the longest side, opposite the right angle
Leg —
either of the two shorter sides of a right triangle
Pythagorean Theorem —
a^2 + b^2 = c^2 for a right triangle
Sine, cosine, tangent —
ratios of a right triangle's sides
Opposite / adjacent —
the leg across from / next to an angle

Why it matters Right-triangle rules turn a tape measure and an angle into distances and heights, used in construction, surveying, and navigation.

Worked examples

Model ItI do

A right triangle has legs a = 6 and b = 8. Find the hypotenuse c.

Use a^2 + b^2 = c^2: 6^2 + 8^2 = 36 + 64 = 100. So c = sqrt(100) = 10.

Try It TogetherWe do

In a right triangle, the angle is 30 degrees, the side opposite it is unknown, and the hypotenuse is 20. Find the opposite side.

Use sine = opposite/hypotenuse. So sin(30) = opposite/20. Since sin(30) = 0.5, the opposite side = 0.5 x 20 = 10.

Now YouYou do

A ladder leans so its foot is 5 ft from a wall and it reaches 12 ft up. How long is the ladder?

The wall and ground form a right angle; use the Pythagorean Theorem. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.In a right triangle, a = 3 and b = 4. Find the hypotenuse c.

  2. 2.Which side of a right triangle is the hypotenuse?

  3. 3.Tangent of an angle equals opposite divided by _____.

On Level

The core Quick Check from the book.

  1. 1.Find c if a = 3 and b = 4 in a right triangle.

  2. 2.Which side is the hypotenuse: a leg or the side opposite the right angle?

  3. 3.Tangent of an angle equals opposite divided by _____.

  4. 4.A right triangle has legs 5 and 12. Find the hypotenuse.

  5. 5.If sin(A) = opposite/hypotenuse and opposite = 7, hypotenuse = 14, find sin(A).

Challenge

Stretch your thinking.

  1. 1.A right triangle has one leg 8 and hypotenuse 17. Find the other leg.

  2. 2.In a right triangle, an angle measures 30 degrees and the hypotenuse is 24. Find the side opposite the 30 degree angle.

  3. 3.A right triangle has legs 9 and 12. Find the hypotenuse.