Determine Missing Side Lengths

How to Determine Missing Side Lengths Using the Pythagorean Theorem

The Pythagorean theorem is one of the most well-known mathematical theorems, and it is particularly important when studying triangles. This article will walk you through the different ways that the Pythagorean theorem may be used, giving you the confidence to handle issues.Determine Missing Side Lengths

The Pythagorean Theorem

A fundamental relationship between the lengths of a right triangle’s three sides is clarified by this theorem. Importantly, only right triangles are covered by the Pythagorean theorem.Determine Missing Side Lengths

A right triangle’s salient characteristics include

• A right angle is made up of one internal angle that is 90 degrees.
• The hypotenuse, or longest side, is always on the other side of the right angle.Determine Missing Side Lengths

The hypotenuse’s square length in a right triangle is equal to the sum of the squares of the other two legs, according to the theorem:

c2 = a2 + b2

In this case, the hypotenuse is denoted by c, while the other two sides are indicated by a and b, which can be used interchangeably.Determine Missing Side Lengths

How a right triangle is identified

Prior to employing the Pythagorean theorem, it is important to make sure that the triangle is a right triangle, since the simplest way to do so is to look for a right angle, which is typically shown by a square symbol.Determine Missing Side Lengths

As an alternative, you can identify a right triangle by using:Determine Missing Side Lengths

• The existence of a right angle is confirmed if two sides are perpendicular, for example, if one is vertical and the other is horizontal.
• Similarly, directional indicators such as North and East/West can indicate perpendicularity.
• Triangle element theorems: For example, if one side is a tangent and the other is a radius, the inherent properties of tangents validate the existence of a right angle.
• Shape properties: Shapes such as squares and rectangles always have right angles, so it is important to take these details into account.Determine Missing Side Lengths

How to Determine Side Lengths That Are Missing

Finding the missing side lengths in a right triangle is the simplest application of the Pythagorean theorem. Let’s examine one example:

Example #1:

Find the missing side of the following right triangle.

The hypotenuse, or longest side, is the one that lacks length.

Step 1:Label the sides, a, b, and c.

For ease of application of the Pythagorean theorem, we designate the right triangle as such, where “c” is always the hypotenuse and a and b can be used interchangeably.

Let a = 5 in and b = 12 in

Step 2: 

Enter the Pythagorean Theorem with the given and unknown numbers.

c² = a² + b²

c²  = 5² + 12²

Step 3: Simplify and solve for c

c²  = 25 + 144

c²  = 169

c = √ 169

c = 13

Thus, 13 inches is the missing side length.

Now let’s examine an example with a real-world setting.

Real-Life Missing Side Length Question

Mike is painting a mural on a tall, vertical wall while utilizing a ladder. He places the ladder one and a half meters from the wall’s base. What is the highest wall Mike can reach if the ladder is five meters long?

Step 1: Verify the right triangle.

The ladder, wall, and floor are the three sides of a right triangle since the angle formed by the vertical wall and the floor is a right angle.

Step 2: Identify the sides

The Pythagorean theorem can be used to determine Mike’s maximum height. According to the theorem, in a right triangle, the square of the hypotenuse’s length—in this example, the ladder—is equal to the sum of the squares of the lengths of the other two sides, which are Mike’s reachable height and the distance between the wall’s base and the ladder.

Step 3: Apply the Pythagorean theorem

c² = a² + b²

Where:

C is the ladder’s length, which is 5 meters.

The distance a between the ladder and the wall’s base is 1.5 meters.

Mike’s maximal height (what we’re looking for) is denoted by b.

Step 4: Replace and figure out b

5² = 1.5² + b²

b² = 5² – 1.5²

b² = 25 – 2.25

b² = 22.75

b = √ 22.75

b = 4.77 (2 d.p.)

Mike can therefore climb to a maximum height of about 4.77 meters on the wall.