right triangles-ritri
Unit outline
ritri 01-simplify radicals, including rationalizing denominator
We know that the square root of 36 (√36) is 6.
However, if we are given √200, we would factor it out to √100 and √2. The square root of √100 is 10 so we could write 10√2.
Also, if we have √72∕√18, we would start off by simplifying 72 and 18 by dividing them by 9 to get √8/√2. Then, we can’t have a square root on the denominator, so we would multiply √8/√2 by √2∕√2 to get √16/2. √16 can be simplified to 4, and 4/2 equals 2. The simplified answer would be 2.
However, if we are given √200, we would factor it out to √100 and √2. The square root of √100 is 10 so we could write 10√2.
Also, if we have √72∕√18, we would start off by simplifying 72 and 18 by dividing them by 9 to get √8/√2. Then, we can’t have a square root on the denominator, so we would multiply √8/√2 by √2∕√2 to get √16/2. √16 can be simplified to 4, and 4/2 equals 2. The simplified answer would be 2.
ritri 02-state pythagorean theorem
The Pythagorean Theorem is:
ritri 03-use pythagorean theorem to solve problems/find missing sides
Example 1:
9² + 12² = x²
81 + 144 = x²
225 = x²
√225 = √x²
15 = x
x = 15
9² + 12² = x²
81 + 144 = x²
225 = x²
√225 = √x²
15 = x
x = 15
Example 2:
5² + x² = 10²
25 + x² = 100
-25 -25
x² = 75
√x² = √75
x = √75
x = √25 · √3
x = 5√3
5² + x² = 10²
25 + x² = 100
-25 -25
x² = 75
√x² = √75
x = √75
x = √25 · √3
x = 5√3
ritri 04-fill in missing side of common pythagorean triplets
Pythagorean triplets are when all 3 sides are integers in a right triangle.
The most common ones to know are 3,4,5 and 5, 12, 13.
Knowing this, we can figure out multiples of these lengths.
For example, if we are given 30, 40, ___, we know that the blank would be 50 because of the 3, 4, 5 triangle.
Also there are two types of formulas to find pythagorean triplets.
One is Plato's formula and it is: n²-1, 2n, n²+1
For example if n=2, the integers would be 3, 4, 5.
The other formula is Euclid's.
It is: 2mn, m²-n², m²+n²
The most common ones to know are 3,4,5 and 5, 12, 13.
Knowing this, we can figure out multiples of these lengths.
For example, if we are given 30, 40, ___, we know that the blank would be 50 because of the 3, 4, 5 triangle.
Also there are two types of formulas to find pythagorean triplets.
One is Plato's formula and it is: n²-1, 2n, n²+1
For example if n=2, the integers would be 3, 4, 5.
The other formula is Euclid's.
It is: 2mn, m²-n², m²+n²
ritri 04h-know and apply euler's formula to find pythagorean triplets
ritri 05-prove pythagorean theorem using area
First, we start off by finding the area of the entire square.
A=l·w In this square, the area is (a+b)(a+b) or a² + 2ab + b². Next, we want to find the area of the 4 triangles, and the smaller square in the center. The area of one triangle would be 1/2·l·w, or or for this triangle 1/2·a·b For the four triangles the area would be multiplied by four, which would be 2ab. The area of the smaller square would be, using our area formula, c². Then, we would set these two areas equal to each other a² + 2ab + b² = 2ab + c² - 2ab - 2ab Subtract 2ab from both sides. Now, you would get a² + b² = c², and your proof is complete. |
ritri 06-determine whether a triangle is right/obtuse or acute
We know that the formula for a right triangle is a² + b² = c².
If the integers of a triangle come up to be a² + b² = c², then the triangle would be right.
If the integers of a triangle come up to be a² + b² < c², then the triangle would be obtuse.
If the integers of a triangle come up to be a² + b² > c², then the triangle would be acute.
*Also keep note that the hypotenuse is always the largest side.
Let's look at some examples:
If the integers of a triangle come up to be a² + b² = c², then the triangle would be right.
If the integers of a triangle come up to be a² + b² < c², then the triangle would be obtuse.
If the integers of a triangle come up to be a² + b² > c², then the triangle would be acute.
*Also keep note that the hypotenuse is always the largest side.
Let's look at some examples:
7² + (2√37)² = √197²
49 + (4·37) = 197 49 + 148 = 197 197 = 197 This triangle is RIGHT because both legs equal the hypotenuse. |
12² + 12² = 15²
144 + 144 = 225 288 = 225 288 > 225 This triangle is ACUTE because the legs are greater than the hypotenuse. |
12² + 8² = 15²
144 + 64 = 225 208 = 225 208 < 225 This triangle is OBTUSE because the legs are less than the hypotenuse. |
ritri 07-state the ratios of sides for special right triangles (30-60-90, 45-45-90)
30-60-90 and 45-45-90 triangles are special because if we are given these angles, we can figure out the side lengths of a triangle. Let's start off with the formula.
ritri 07h-derive the ratios of sides for special right triangles
If you are given the lengths b or c in these triangles, you can derive it algebraically with this formula.
ritri 08-use special right triangles to solve problems
Using our above formulas, let's solve right triangle problems.
ritri 09-find the area of regular polygon, given apothem AND side
First of all, to find the area of a polygon, you need to know 3 items.
The first is the number (n) of sides of the polygon you are looking at.
The second is the length of one of the sides (s) of your regular polygon.
And the third is the apothem (a). The apothem is a line segment from the center of a regular polygon to the midpoint of a side.
The formula for any regular polygon is this:
A(n-gon) = ½·a·s·n
Let's look at an example:
The first is the number (n) of sides of the polygon you are looking at.
The second is the length of one of the sides (s) of your regular polygon.
And the third is the apothem (a). The apothem is a line segment from the center of a regular polygon to the midpoint of a side.
The formula for any regular polygon is this:
A(n-gon) = ½·a·s·n
Let's look at an example:
ritri 10-find the area of a regular polygon, given apothem or side. (tri, quad, hex)
Using the same formula and steps as above, we could solve the same problem if given only the apothem OR the side.
Let's look at an example:
Let's look at an example:
We are given that 8 is the apothem. But we need to find the length of a side. This requires us using skills about special right triangles.
We could draw a radius going from the center point to the vertex, bisecting that angle. Since this is a hexagon, each interior angle would equal 120, and since the angle is bisected, the interior angle is 60. This now gives us a 30-60-90 triangle. Now we just have to solve to find the side. 8 * 2=16 16√3/3 Now we can plug it in to the formula. a=8 s=16√3/3 n=6 A=½·8·16√3/3·6 This would come out to be: A=768√3/6 which simplifies to 128√3 |
ritri 11-derive the distance formula from scratch
ritri 12-use the distance formula, simplify result, to find distance
Example:
Find the distance between (3, 2) and (7, 8).
Plug these numbers into the distance formula.
Find the distance between (3, 2) and (7, 8).
Plug these numbers into the distance formula.
So the answer is 2√15.
ritri 13-write the equation for a circle, given center and radius. (and vice versa)
Example:
Write the formula for a circle given that the center is (2, -3) and the radius is 2√7
Write the formula for a circle given that the center is (2, -3) and the radius is 2√7