GOSEA 8 - Pythagorean Theorem

GOSEA 8

Pythagorean Theorem

Named after the Greek mathematician Pythagoras.

c. 570 - c.495 BC

What is a theorem?

In math, there are a lot of different kinds of rules with different kinds of names. Postulates and theorems are kinds of geometrical rules.

Postulates can not be proven to be true; they are simply assumed to be true.

Theorems can be proven to be true.

These statements will make more sense when we start actually proving theorems in the latter part of Unit 2.

Vocabulary needed for this lesson

-right angle

-triangle

-right triangle

-leg of a right triangle

-hypotenuse

Right angle

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A right angle measures exactly 90° .

A sheet of notebook paper has a right angle at each corner. 

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Triangle

A triangle is a geometric figure that has three sides and three angles.

The prefix "tri" means three.

Right triangle

A right triangle is a triangle that has exactly one right angle.

A right triangle can not have more than one right angle.

Therefore, the remaining two angles are smaller than right angles. (Do you remember what these two smaller angles are called? Answer is on next slide.)

Leg of a right triangle

Every triangle has three sides.

In a right triangle, the two sides that come together to form the right angle are called the "legs" of the triangle.

(The answer to the question on the previous slide is "acute angles".)

Hypotenuse

When a triangle has a hypotenuse, we automatically know the triangle is a right triangle.

The hypotenuse is ALWAYS the longest side and is always across from the right angle.

The hypotenuse will be a side to BOTH acute angles in a right triangle.​

Multiple Select

Which two sides appear to form a right angle?

• a. pink

• b. blue

• c. green

Multiple Choice

Which side is across from the right angle?

• a. pink

• b. blue

• c. green

Multiple Choice

Which side is the "short leg"?

• a. pink

• b. blue

• c. green

Multiple Choice

Which side is the "long leg"?

• a. pink

• b. blue

• c. green

Multiple Choice

Which side is the "hypotenuse"?

• a. pink

• b. blue

• c. green

Pythagorean Theorem

In a right triangle (and only a right triangle), the square of one leg plus the square of the other leg equals the square of the hypotenuse.

Pythagorean Theorem

Squaring a number means to multiply it together twice.

Squaring a side means to make a square using the given side as the length of all sides of the square.

Pythagorean Theorem

In the given triangle, a and b are the legs and c is the hypotenuse of the right triangle.

Pythagorean Theorem

In a right triangle where

a and b are the legs and

c is the hypotenuse,

then a² + b² = c².

Pythagorean Theorem

Pythagorean Theorem

Don't be scared of something new.

Some problems require formulas.

When a problem requires a formula to solve it, it is ALWAYS a good idea to write the formula as the first step.

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Doing so could earn you some partial credit on written assignments even if your work is incorrect (as long as you write the correct formula and write it correctly).

So when do we use the Pythagorean Theorem anyway?

The Pythagorean Theorem can only be used with right triangles.

It is most often used when two side lengths are known and the third side length must be found.

So when do we use the Pythagorean Theorem anyway? (continued)

A less popular use of the Pythagorean Theorem is to use it to check to see if a triangle is a right triangle or not.

We will start with these kinds of problems.

Multiple Choice

Is this triangle a right triangle? Does the leg squared plus the other leg squared equal the hypotenuse squared?

• The triangle is a right triangle because $$10^2+6^2=8^2$$

• The triangle is a right triangle because $$8^2+6^2=10^2$$

• The triangle is not a right triangle because $$6^2+8^2<10^2$$

• The triangle is not a right triangle because $$6^2+8^2>10^2$$

Multiple Choice

Is this triangle a right triangle? 

• The triangle is a right triangle because $$10^2+9^2=15^2$$

• The triangle is a right triangle because $$9^2+15^2=10^2$$

• The triangle is not a right triangle because $$9^2+10^2>15^2$$

• The triangle is not a right triangle because $$9^2+10^2<15^2$$

Multiple Choice

Is this triangle a right triangle? Does the leg squared plus the other leg squared equal the hypotenuse squared?

• The triangle is a right triangle because $$10^2+12^2=15^2$$

• The triangle is a right triangle because $$12^2+15^2=10^2$$

• The triangle is not a right triangle because $$12^2+10^2<15^2$$

• The triangle is not a right triangle because $$12^2+10^2>15^2$$

Multiple Choice

Can the following three sides form a right triangle?

Yes or no? Why?

3cm, 4cm, 5cm

• No;

  $$3^2+4^2=5^2$$  

• Yes; $$3^2+4^2=5^2$$

• No; $$3^2+4^2<5^2$$

• Yes; $$3^2+4^2>5^2$$

Multiple Choice

Can the following three sides form a right triangle?

Yes or no? Why?

7cm, 12cm, 13cm

• No;

  $$7^2+12^2>13^2$$

• Yes; $$7^2+12^2=13^2$$

• No; $$7^2+12^2<13^2$$

• Yes; $$7^2+12^2>13^2$$

The other use for the Pythagorean Theorem...

In Geometry, we will primarily use the Pythagorean Theorem to find the missing side of a RIGHT TRIANGLE.

The Pythagorean Theorem CAN NOT be used on a triangle that is not a right triangle.

Multiple Choice

Which is the correct equation for finding the missing side of the right triangle?

• $$x^2+6^2=8^2$$

• $$x^2+8^2=6^2$$

• $$6+8=x^2$$

• $$6^2+8^2=x^2$$

Multiple Choice

Which is the correct equation for finding the missing side of the triangle?

• $$x^2+12^2=15^2$$

• $$x^2+15^2=12^2$$

• $$12^2+15^2=x^2$$

• $$x^2+12=15$$

Multiple Choice

Which is the correct equation for finding the missing side of the triangle?

• $$x^2+5=12$$

• $$x^2+12^2=5^2$$

• $$12^2+5^2=x^2$$

• $$x^2+5^2=12^2$$

Multiple Choice

Which is the correct equation for finding the missing side of the triangle?

• $$x^2+5^2=3^2$$

• $$x^2+3^2=5^2$$

• $$3^2+5^2=x^2$$

• $$x^2+3=5$$

Multiple Choice

Choose the correct equation for finding the missing side

of the right triangle with the given sides.

Side c is the hypotenuse. Sides a and b are the legs.

b = 8 km, c = 10 km

• $$8^2+10^2=a^2$$

• $$a^2+8^2=10^2$$

• $$a^2+10^2=8^2$$

Multiple Choice

Choose the correct equation for finding the missing side

of the right triangle with the given sides.

Side c is the hypotenuse. Sides a and b are the legs.

a = 5 m, b  = 12 m

• $$5^2+12^2=c^2$$

• $$a^2+5^2=12^2$$

• $$a^2+12^2=5^2$$

Some things you should know...

• x will not always be the variable used when finding a missing side.

• You will need to use whatever variable is in the problem.

  Pick your own variable only when one is not already being used.

• You will have to solve these equations.

• You will have to show work when solving these equations.

• You will be taught HOW to show the work.

• Mrs. Haley's University students may be required to show work using units. This will become helpful when taking Chemistry.

Keep doing this assignment until you make a 100. That is the only way you will get the 100 for Geometry in the Fall.​