Naming the Legs and Hypotenuse of a Right Triangle | Grade 8 Math | LevelKite

A right angle is an angle that measures exactly $90^\circ$ , like the corner of a sheet of paper. A right triangle is a triangle that has a right angle.

The three sides of a right triangle have special names. The good news is that every name depends on one thing: where the right angle is. So in this lesson you will find the right angle first and then name the sides from it.

Start with the triangle in the figure below. In every figure in this lesson, the horizontal axis is the $x$ -axis, the vertical axis is the $y$ -axis, and each grid square is 1 unit wide and 1 unit tall.

This triangle has vertices at

(1, 1)

,

(5, 1)

and

(1, 4)

. The small square at

(1, 1)

marks its right angle.

Each corner of a triangle is a vertex (plural: vertices). The triangle in the figure has vertices at $(1, 1)$ , $(5, 1)$ and $(1, 4)$ . Use these coordinates, not the look of the drawing, to find the right angle:

$(1, 1)$ and $(5, 1)$ have the same $y$ -coordinate, $1$ , so the side that joins them is horizontal.
$(1, 1)$ and $(1, 4)$ have the same $x$ -coordinate, $1$ , so the side that joins them is vertical.

On the coordinate plane, a horizontal line and a vertical line always cross at a right angle. These two sides meet at $(1, 1)$ , so the right angle of this triangle is at $(1, 1)$ . The small square drawn in that corner is the usual mark for a right angle. When a drawing carries that mark, it is telling you that the angle is exactly $90^\circ$ .

Now name the sides. The two sides that meet at $(1, 1)$ form the right angle. They are the legs of the right triangle: the horizontal side from $(1, 1)$ to $(5, 1)$ and the vertical side from $(1, 1)$ to $(1, 4)$ .

The third side runs from $(5, 1)$ to $(1, 4)$ and joins the far ends of the legs. It is the only side that does not touch the vertex $(1, 1)$ , so it faces the right angle from across the triangle. A side that faces an angle in this way is opposite that angle. The side opposite the right angle is the hypotenuse.

Key concept

Legs and hypotenuse of a right triangle

In a right triangle, the right angle decides the name of every side:

The legs are the two sides that form the right angle. They meet at its vertex.
The hypotenuse is the side opposite the right angle. It is the one side that does not touch the vertex of the right angle.

The right angle defines these names, so they apply only to right triangles. A triangle with no right angle has no hypotenuse. Always find the right angle first, from coordinates, a right-angle mark or a statement that the angle is $90^\circ$ , and never from the way a corner looks.

Triangle

ABC

: Point A is at

(2, 6)

, Point B is at

(5, 6)

and Point C is at

(5, 1)

.

Worked example

Name the sides from coordinates

Triangle $ABC$ in the figure has vertices Point A at $(2, 6)$ , Point B at $(5, 6)$ and Point C at $(5, 1)$ . Side $AB$ means the side that joins Point A and Point B. The drawing has no right-angle mark. Which sides are the legs, and which side is the hypotenuse?

Look for a horizontal side. Point A and Point B have the same y-coordinate, 6, so side AB is horizontal.
$A(2, 6) \text{ and } B(5, 6)$
Look for a vertical side. Point B and Point C have the same x-coordinate, 5, so side BC is vertical.
$B(5, 6) \text{ and } C(5, 1)$
Locate the right angle. On the coordinate plane a horizontal line and a vertical line cross at a right angle, and sides AB and BC meet at Point B, so the right angle is at Point B.
$\angle B = 90^\circ$
Name the legs. The legs are the two sides that form the right angle, so they are the two sides that meet at Point B.
$\text{legs: } AB \text{ and } BC$
Name the hypotenuse. It is the side opposite the right angle, the only side that does not touch Point B. That side joins Point A and Point C.
$\text{hypotenuse: } AC$

Answer

In triangle $ABC$ , the legs are sides $AB$ and $BC$ , and the hypotenuse is side $AC$ .

Triangle

DEF

: Point D is at

(1, 1)

, Point E is at

(2, 3)

and Point F is at

(6, 1)

. The small square at Point E marks the right angle.

Worked example

Name the sides when the legs are slanted

Triangle $DEF$ in the figure has vertices Point D at $(1, 1)$ , Point E at $(2, 3)$ and Point F at $(6, 1)$ . The angle at Point E is a right angle, and the small square at Point E marks it. Which sides are the legs, and which side is the hypotenuse?

Find the right angle. No two vertices share an x-coordinate, so this triangle has no vertical side and you cannot find the right angle the way you did in triangle ABC. Use the statement in the problem instead: the right angle is at Point E.
$\angle E = 90^\circ$
Name the legs. The legs form the right angle, so they are the two sides that meet at Point E. Each one joins two points that differ in both coordinates, so each is slanted, meaning neither horizontal nor vertical. That does not change its name.
$\text{legs: } DE \text{ and } EF$
Name the hypotenuse. It is the side opposite the right angle, the only side that does not touch Point E. That side joins Point D and Point F.
$\text{hypotenuse: } DF$
Check the direction of the hypotenuse from the coordinates. Point D and Point F have the same y-coordinate, 1, so side DF is horizontal.
$D(1, 1) \text{ and } F(6, 1)$

Answer

In triangle $DEF$ , the legs are the slanted sides $DE$ and $EF$ , and the hypotenuse is the horizontal side $DF$ .

One more fact gives you a quick check. In a right triangle, the hypotenuse is the longest side.

Here is the reason. The three angles of any triangle add up to $180^\circ$ . In a right triangle the right angle uses $90^\circ$ , so the other two angles share the remaining $90^\circ$ and each of them is smaller than $90^\circ$ . That makes the right angle the largest angle. In any triangle, a larger angle has a longer side opposite it. So the side opposite the right angle, the hypotenuse, is longer than each leg.

Use this check only when you already know that the triangle is a right triangle. A triangle with no right angle can still have a longest side, but it has no hypotenuse. So a longest side, by itself, never shows that a triangle has a right angle.

Worked example

Name the sides from a description

A ladder leans against a wall. The wall is vertical and the ground is level, so the wall and the ground meet at a right angle. The foot of the ladder is 5 feet from the wall, the top of the ladder touches the wall 12 feet above the ground, and the ladder is 13 feet long. In the triangle formed by the ladder, the wall and the ground, which sides are the legs and which side is the hypotenuse?

List the three sides with their lengths so that you can match each name to a length: the stretch of ground from the wall to the foot of the ladder is 5 feet, the stretch of wall from the ground up to the top of the ladder is 12 feet, and the ladder itself is 13 feet.
Find the right angle. The problem states that the wall and the ground meet at a right angle, so the right angle is at the corner where the wall meets the ground.
Name the legs. The two sides that meet at that corner are the 5-foot side along the ground and the 12-foot side up the wall, so these two sides form the right angle.
$\text{legs: } 5 \text{ ft and } 12 \text{ ft}$
Name the hypotenuse. The ladder joins the ground to the wall without touching the corner, so it is the side opposite the right angle.
$\text{hypotenuse: } 13 \text{ ft}$
Check with the longest-side fact. It applies because this is a right triangle, and it says the hypotenuse must be longer than each leg.
$13 > 12 \text{ and } 13 > 5$

Answer

The legs are the 5-foot side along the ground and the 12-foot side up the wall, and the hypotenuse is the 13-foot ladder.

Takeaway: In a right triangle, find the right angle first. The two sides that form it are the legs. The side opposite it is the hypotenuse, and it is also the longest side. Find the right angle from coordinates, a right-angle mark or a clear statement, and never name a side by the way it points.

Next, you will use these names to find the hypotenuse or a missing leg of a right triangle. Call the lengths of the legs $a$ and $b$ and the length of the hypotenuse $c$ , all positive numbers in the same unit. The rule you will use, $a^2 + b^2 = c^2$ , applies only to a right triangle, and only when $c$ is the hypotenuse, the side opposite the right angle. For the ladder, that means $a$ and $b$ are $5$ and $12$ and $c$ is $13$ , all in feet.