Right Triangles, Pythagorean Theorem & Distance Formula

Right Triangles, Pythagorean Theorem & Distance Formula

to be used with accompanying wehewehe doc

Sides of a Right Triangle

The hypotenuse is the longest side, it is also opposite the right Angle.

The Legs are the other two sides.

Angles & Sides- Its Relative!

To distinguish one leg from the other, we use the terms Opposite and Adjacent, but it depends on the Angle we are referring to. Look at the diagram and read the following

Angle A is Opposite side a and...

Angle A is Adjacent to side b

Angle B is Opposite side b and...

Angle B is Adjacent side a

Angle C is opposite side c, but we don't call it opposite because c is the hypotenuse

Multiple Choice

Who is is legendary Ionian Mathematician /Philosopher?

• Osama bid Laden

• Fidel Castro

• Pythagoras

• Rasputin

• Leonard Da Vinci

Multiple Choice

Which of the following is a right triangle?

Multiple Choice

What do we call the side across from the right angle?

• Leg

• da kine

• Hypotenuse

• Obtuse

• Adjacent

Multiple Choice

What is always true about the hypotenuse?

• Is the shortest side of the triangle.

• Is the height of the triangle.

• Is labeled as b.

• Is the longest side of the triangle.

Multiple Choice

Given a right triangle ABC, name the hypotenuse.

• CB

• AC

• AB

• ab

Multiple Choice

From ∠C, what side is AB?

• Adjacent

• Hypotenuse

• Opposite

Multiple Choice

From ∠C, what side is BC?

• Adjacent

• Hypotenuse

• Opposite

The Pythagorean Theorem

Proving The Pythagorean Theorem

Time to intereact to understand what the theorem says and how we know its always true for Right Triangles...use links on doc

Multiple Choice

All of the green "a" square plus all of the purple "b" square will perfectly fill the red "c" square.

• True

• False

• who could know this?

Multiple Choice

If the side lengths of a right triangle are 7, 25, and 24, which side is the hypotenuse?

• 7

• 25

• 24

• none of the these

Multiple Choice

If the side lengths of a right triangle are 7, 25, and 24, which side squared equals the sum of the other two squared?

• 7

• 25

• 24

• none of the these

Multiple Choice

—State the Pythagoras Theorem using the information given in the triangle.

• w² + y² = x²

• y + x = w

• w + x = y

• w² + x² = y²

The Converse of the Pythagorean Theorem

Is it Right?

To use the Converse of the PT, put all three side lengths into the PT correctly and check if both sides are equal.

Multiple Choice

a=6, b=8, and c=10

Is this a right triangle?

• yes

• no

• not enough information to tell, need to know angles...

Multiple Select

Which side length triples would form a right triangle?

• 5, 12, 13

• 2, 8, 10

• 30, 40, 50

• 7, 12, 15

Multiple Choice

Does an 8, 15, 16 triangle have a Right Angle?

• Yes

• no

Pythagorean Triples

It turns out there are many side length(whole number) triples that form Right Triangles. The 3,4,5 is the smallest of these Pythagorean Triples and is called "primative". We know that if we multiply all the sides of that triangle by a scale factor, the new triangle is similar and therefore also a Pythagorean Triple... These are said to be the same family of Pythagorean Triples.

Some Families of Pythagorean Triples

...notice that each "daughter" triple is similar to it "parent".

Multiple Choice

Does the set of numbers represent a Pythagorean Triple.

4, 5, 6

• Yes

• No

Multiple Choice

Which is a Pythagorean Triple?

• 4, 4, 8

• 6, 8, 15

• 10, 24, 26

• 10, 15, 25

Solving for the Hypotenuse

What is a square root?

Square Roots & Right Triangles

This spiral is made by starting with an Isosceles Right Triangle with sides of 1. The hypotenuse of that triangle becomes the leg of the next, and another side of 1 is used to make the next RT...It happens that the sequence of Hypotenuses are then..... make one for extra credit

Multiple Choice

What is the opposite of squaring a number?

• divide

• multiply

• fractions

• square root

Multiple Choice

solve for c

• 7

• 9

• 4

• 13

Multiple Choice

What is the length of x? (click on picture)

• 5.7

• 10.6

• 20

• 400

Multiple Choice

What is the length of the hypotenuse?

• 33.5 cm
• 45 cm
• 6.7 cm
• 15 cm

Solving for a Leg

Multiple Choice

^{Which equation would you use to find the missing side?}

• 48² + x² = 60²

• 48² - x² = 60²

• 48² + 60² = x²

• 60² + x² = 48²

Multiple Select

Given this Right triangle, which of the following statements are TRUE?

• w² + y² = x²

•  $x=\sqrt{y^2-w^2}$  

•  $y=\sqrt{x^2+w^2}$  
  

• w² + x² = y²

Multiple Choice

Evaluate x.

• 35

• 20

• 8

• 12

Multiple Choice

^{Find the missing side}

• 9

• 26.9

• 37

• 18.2

Applying the Pythagorean Theorem

The Pythagorean Theorem is one of the most useful of all Geometry Concepts. Lets look at a few examples....

Diagonals of Rectangles

Since Rectangles (and squares) have right angles, we can find the diagonal using the PT...

Multiple Choice

A square has an area of 64 square cm. What is the lenght of its diagonal?

• 8cm

• 16cm

• 10cm

• 11.31cm

• 12.2cm

Multiple Choice

What is the area of this Rectangle?

• 15 cm²

• 6 cm²

• 34 cm²

• 12 cm²

• 24 cm²

Multiple Choice

Casey travels from her house directly west to the bank and then directly north from the bank to the mall. She then travels home on the road connecting the mall and her house. What is her total distance traveled?

• 17 miles
• 32 miles
• 37 miles
• 40 miles

Multiple Choice

Find h

• 35.60
• 35.00
• 35.71
• 61.03

Multiple Choice

A piece of paper that Brittany has is 11 inches tall and 8 inches wide. She draws a straight line diagonally across the paper. How long is the line she drew?

• 7.5 in

• 12.4 in

• 13.6 in

• 14.3 in

Multiple Choice

A piece of paper that Brittany has is 11 inches tall and 8 inches wide. She draws a straight line diagonally across the paper. How long is the line she drew?

• 7.5 in

• 12.4 in

• 13.6 in

• 14.3 in

The Distance Formula

The distance between any two points can be found using the PT. When we solve for d (really just the hypotenuse length) we call it the distance formula. Here it is what it looks like for two (x,y) coordinates.

Multiple Choice

How is the distance formula correctly written:

• $d\ =\ \sqrt{\left(y_{1\ }-\ y_2\right)^2\ -\left(x_{2\ }-\ x_1\right)^2}\$

• $d\ =\ \sqrt{\left(x_{2\ }-\ x_1\right)^2\ +\ \left(y_{2_{\ }}-\ y_1\ \right)2}$

• $d\ =\ \sqrt{\left(y_{1\ }-\ y\ _2\right)^2\ +\ \left(x_{2\ }-\ x_1\right)}^2$

• d = $\sqrt{\left(x\right)^2-\ \left(y\right)^2}$

Multiple Choice

How is AB written in the distance formula?

• 
  $d\ =\ \sqrt{\left(5-3\right)^2\ +\ \left(0--2\right)^2}$
  

• $d\ =\ \sqrt{\left(2-5\right)^2\ +\ \left(3\ -0\right)^2}$

• $d\ =\ \sqrt{\left(5-3\right)^2\ -\ \left(0--2\right)^2}$

• $d\ =\ \sqrt{\left(4-3\right)^2\ -\ \left(2-4\right)^2}$

Multiple Choice

A(2,0) B(-2,4) is written as

• $d\ =\ \sqrt{\left(4-2\right)^2\ -\ \left(0-2\right)^2}$

• $d\ =\ \sqrt{\left(4-0\right)^2\ -\ \left(-2-2\right)^2}$

• $d\ =\ \sqrt{\left(-2\ +0\right)^2\ +\ \left(4+2\right)^2}$

• $d\ =\ \sqrt{\left(-2-2\right)^2\ +\ \left(4-0\right)^2}$

Multiple Choice

A(3,1) B(-2,-1) written correctly is:

• $\sqrt{\left(-2-3\right)^2\ +\ \left(-1-1\right)^2}$

• $\sqrt{\left(1-3\right)^2\ +\ \left(-1-2\right)^2}$

• $\sqrt{\left(-2-3\right)^2\ -\ \left(-1-1\right)^2}$

• $\sqrt{\left(-2-3\right)^2\ -\ \left(-1-1\right)^2}$

Multiple Choice

Find the distance between ( 1, 3) and (5, 6)

• 3

• 4

• 5

• 6

Multiple Choice

Find the distance between J and K.

• 8.2

• 7.7

• 9

• 10.2

Multiple Choice

Find the direct distance between the court house and the town hall.

• 26

• 6.8

• 10

• 7.2 km

Multiple Choice

AB = 5 on the coordinate plane shown. If B(2,1), which point could NOT be another location of A?

• (-1,5)

• (6,4)

• (-2,-2)

• (4,-3)

• (-3,1)

This concludes our Lesson on Right Triangles, The pythagorean Theorem and The distance Formula. Next you will be learning about.....

The PT in 3D... and

The Trianlge Inequality Theorem

Which helps us know if a Triangle is Right, Acute or Obtuse. We will also use similarity to work with...

Special Right Triangles

...which we will use to develop the basis for..

Trigonometric Ratios

Which will finally allow us to relate sides and angles.....and if all goes well, also..

Trig Functions

...but not until after Spring Break!

Poll

Rate your Confidence in Understanding what you have learned using this Lesson and the wehe links...

• I think i got it all

• mostly good, a few questions still

• ummm, at least i tried

• what just happened?