Chapter 9

Perimeter of a rhombus

- P = 4s - Like any polygon, the perimeter is the total distance around the outside, which can be found by adding together the length of each side. - In the case of a rhombus, all four sides are the same length by definition, so the perimeter is four times the length of a side

hypotenuse

- The hypotenuse is always opposite the right angle and it is always the longest side of the triangle. - To find the length of leg 1 and 2, substitute the known values into the Pythagorean Theorem. Solve for x2. - (leg1)2 + (leg2)2 = (hypotenuse)2

Pythagorean triple Examples

- 3 4 5 - 5 12 13 - 8 15 17 - 7 24 25

30 degree - 60 degree - 90 degree Triangle Theorem

- 30-60-90 triangle - find the measure of any of the three sides, simply by knowing the measure of at least one side in the triangle. - The hypotenuse is equal to twice the length of the shorter leg, which is the side across from the 30 degree angle.

45 degree - 45 degree - 90 degree Triangle Theorem

- 45-45-90 triangle means a triangle with two 45 degree angles and one 90 degree angle. - A 45-45-90 triangle has two sides that are of equal length, called the legs. - The third side is longer than the other two and is called the hypotenuse and is always opposite the right angle - sr 2 * x or x * sr 2 - hypotenuse = sr 2 * leg; then hypotenuse/sr 2 = leg

Is the triangle with side lengths 7, 8, 9 acute, obtuse or right?

- 49 + 64 = 81 - 113 > 81 - acute, c2< a2 + b2

Rhombus

- A 4-sided flat shape with straight sides where all sides have equal length. - Also opposite sides are parallel and opposite angles are equal. It is a type of parallelogram.

Altitude of an Equilateral Triangle

- An equilateral triangle is a special case of a triangle where all 3 sides have equal length and all 3 angles are equal to 60 degrees.

If c2 > a2+b2, where c is the longest side of a triangle, then the triangle is acute.

- False - Obtuse triangle

Pythagorean Theorem can be used for any triangle?

- False - Only on right triangles - But the theorem can help identify acute and obtuse triangles

Converse of the Pythagorean Theorem

- If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

tangent

- In a right triangle, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side

Pythagorean triple

- It is perhaps surprising that there are some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles. The three whole number side-lengths are called a Pythagorean triple or triad. An example is a = 3, b = 4 and h = 5, called "the 3-4-5 triangle".

Does 4, 5, 6 form a Pythagorean triple?

- No - 16 + 25 > 36, 41 > 36 - Acute, c2 < a2 + b2

trigonometric ratios

- The six trigonometric ratios relate the sides of a right triangle to its angles. Specifically, they are ratios of two sides of a right triangle and a related angle. - Trigonometric functions are typically used to calculate unknown lengths or angles in a right triangle - if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle - because these similar right triangles have equal ratios, the are trigonometric ratios: sine, cosine, and tangent

Diagonal of a rectangle

- Use the Pythagorean Theorem - Could be a triple

Diagonal of a square

- Use the Pythagorean Theorem - Could be a triple - To find the length of the diagonal of a square, multiply the length of one side by the square root of 2: If the length of one side is x... length of diagonal = x. The central angle of a square: The diagonals of a square intersect (cross) in a 90 degree angle.

This triangle is a right triangle: 9, 12, 15

- Yes - 81 + 144 = 225

Does 10, 24, 26 form a Pythagorean triple?

- Yes - 100 + 576 = 676 - Right, c2 = a2 + b2

Rectangle

- a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). - It can also be defined as a parallelogram containing a right angle

Pythagorean triple

- a set of positive integers a, b, and c that satisfy that equation a2+b2=c2. - this equation c must be the greatest number. - if you have a right triangle, then the lengths of the sides will form a Pythagorean triple

Pythagorean Triple

- a set of three positive integers a, b, and c that satisfy the equation a2 + b2 = c2 is called a Pythagorean Triple

Square

- a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). - It can also be defined as a rectangle in which two adjacent sides have equal length.

Altitude

- an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e. forming a right angle with) a line containing the base (the opposite side of the triangle). - This line containing the opposite side is called the extended base of the altitude

angle of elevation

- an angle formed by a horizontal line and the line of sight to an object that is above the horizontal line

angle of depression

- an angle formed by a horizontal line and the line of sight to an object that is below the horizontal line

Equilateral Triangle

- an equilateral triangle is a triangle in which all three sides are equal

Isosceles Triangle

- an isosceles triangle is a triangle that has two sides of equal length. - Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.

elevation and depression

- angles of elevation and depression as the acute angles of right triangle formed by a horizontal distance and a vertical height

finding distance of a square with diagonal

- distance or hypotenuse = sr 2 * leg

Pythagorean Theorem

- if a triangle is a right triangle, then the sum of the squares of the lenghts of the legs is equal to the square of the length of the hypotenuse. - a property of all triangles with a right-angle (an angle of 90°)

Obtuse Triangle

- if the square of the length of the longest side of the triangle is greater than the sum of the squares of the lengths of the other two sides, that the triangle is obtuse - c2 > a2 + b2

Acute Triangle

- if the square of the length of the longest side of the triangle is less than the sum of the squares of the lengths of the other sides, then the triangle is acute - c2 < a2 + b2

Right Triangle

- if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. - c2 = a2 + b2

solving the triangle

- process of determining the three angles and the lengths of the three sides of a triangle - can solve if you know 1) lengths of two sides or length of one side and the measure of one acute angle - one angle is a 90 (right) angle - use Pythagorean Theorem - use trigonometric ratios

Hinge Theorem

- states that the longer side is opposite the larger angle and the shorter side is opposite the smaller angle - determining acute or obtuse

cosine

- the trigonometric function that is equal to the ratio of the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse.

sine

- the trigonometric function that is equal to the ratio of the side opposite a given angle (in a right triangle) to the hypotenuse

positive integers

1, 2, 3, 4, ...