# What is a Right Triangle

## Right triangle definition

All triangles have interior angles adding to **180°**. When one of those interior angles measures **90°**, it is a **right angle** and the triangle is a **right triangle**. In drawing right triangles, the interior **90°** angle is indicated with a little square **□** in the vertex.

The term "right" triangle may mislead you to think "left" or "wrong" triangles exist; they do not. "Right" refers to the Latin word *rectus*, meaning "upright."

## Hypotenuse and sides of a right triangle

We already know the square vertex of the right triangle is a right angle. Opposite it is the triangle's **hypotenuse**, the longest of the three sides, usually labeled * c*.

The other two angles in a right triangle add to **90°**; they are complementary. The **sides** opposite the **complementary angles** are the triangle's legs and are usually labeled * a* and

*.*

**b**## Properties of right triangles

## Construct a right angled triangle

Use two uncooked spaghetti strands to make your own right triangle. Leave one alone; break the other strand into two unequal lengths.

Place the two short strands * a* and

*so they meet at two endpoints and form a*

**b****90°**angle. Laying the third strand

*down to intersect the two endpoints of aa and bb creates a right triangle.*

**c**You can make a more accurate right triangle by using graph paper and a straightedge. Draw a line segment (of any desired length) along the graph paper's printed lines. Follow the lines to make a second line segment exactly **90°** to your first line segment, of any desired length.

If you connect the two endpoints of those line segments, you have a right triangle.

Geometry uses symbols as shorthand. Here are important ones to know:

**∼**means "similar"**∠**means "angle"**△**means "triangle"

## The Pythagorean theorem

Greek mathematician Pythagoras gets the credit, but other civilizations knew about this theorem. The **Pythagorean Theorem** describes the relationship between the lengths of legs * a* and

*of any right triangle to the length of hypotenuse,*

**b***:*

**c**The sum of the squares of legs, * a* and

**b***,*are equal to the square of hypotenuse,

*, or*

**c**Thousands of proofs of this theorem exist, including one by U.S. president James Garfield (before he became president). One proof is easy to make with graph paper, a straightedge, pencil, and scissors.

Construct **△ABC** with legs, * a* and

*, to the left and bottom and hypotenuse,*

**b***at the top right. Leg*

**c***is opposite*

**a****∠A**, leg bb is opposite

**∠B**, and hypotenuse cc is opposite right angle

**C**.

Let length **a=3**, **b=4**, and hypotenuse** c=5**.

Construct a square using leg aa as the right side of the square. It will be** 9** square units (${a}^{2}$). Construct a square using leg * b* as the top side of its square, so it is

**16**square units (${b}^{2}$). Cut out another

**5 x 5**square and line it up with hypotenuse,

*, so the square is ${c}^{2}$.*

**c**Think: what is **9** square units + **16** square units? It is **25** square units, the area of ${c}^{2}$.

## Right triangle altitude theorem

The right triangle altitude theorem tells us that the altitude of a right triangle drawn to the hypotenuse * c* forms two similar right triangles that are also similar to the original right triangle.

Construct **△ABC** so that hypotenuse * c* is horizontal and opposite right angle

**C**, meaning legs aa and bb are intersecting above

*to form the right angle*

**c****C**. This puts

**∠A**to the bottom left, and

**∠B**to the bottom right.

Construct an altitude (or height) * h* from the interior right angle

**C**to hypotenuse

*(so it is perpendicular to*

**c***).*

**c**This altitude * h* creates two smaller triangles inside our original triangle. The altitude divided

**∠C**, and also created two right angles where it intersected hypotenuse

*.*

**c**Call the point where the altitude * h* touches hypotenuse,

**c***,*point

**D**. You now have two triangles,

**△ACD**and

**△BCD**. Each of these triangles is similar to the other triangle, and both are similar to the original triangle.

You can prove this by seeing that new triangle's **∠ADC = original triangle's ∠ACB**, while new triangle's **∠CAD = original triangle's ∠CAB**.

This means two angles of **△ADC** and **△ABC** are similar, making the triangles themselves similar (by the Angle-Angle postulate or AA postulate):

Go through the figure again, concentrating on the larger, new triangle **BCD**. Here **∠BDC = ∠ACB**, and **∠DBC = ∠ABC**, so again, (by the AA postulate):

Since each of the two smaller triangles are similar to the larger triangle, they are similar to each other. So:

## Lesson summary

After going through the videos, reading the lesson and examining the pictures, you now know how to identify a right triangle (by its interior right angle), what its identifying property is (it has one interior right angle).

You also know what the Pythagorean Theorem is (${a}^{2}+{b}^{2}={c}^{2}$) and how to prove it, and what the right triangle altitude theorem is (the altitude of a right triangle drawn to the hypotenuse, * c*, forms two similar right triangles that are also similar to the original right triangle) and how to prove it.