9.3 - apply the Pythagorean theorem and it's Converse

9.3- Semicircles are constructed on each side of triangle△ ABC attached. <C is a right triangle with AC = 6 units and BC = 8 units. What is the sum of the areas of the semicircles?

By the pythagorean theorem AB equals 10 units thus the some of the areas of the semi circles is given by: ½π(3)²+½π(4)²+½π(5)²=25π Competency 9- use the properties of polygons to solve problems.

Hypotenuse Leg Theorem

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Converse of the Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.

Isosceles Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Rectangle, square, quadrilateral, parallelogram and rombus

It has 4 sides so it is a quadrilateral. This quadrilateral has all right angles so it is a rectangle. A rectangle has 4 sides equal so that we can say it to rhombus. this rhombus has all angles 90ﾟ so it's a square. every square is a parallelogram

Arc Length Formula

L = 2πr (m⁰/360⁰), radians s=rθ

the base of the pyramid is a square with side lengths of 30 in. The height of the pyramid is 50 inches. Find the slant height

Slant height of a square Pyramid: s=√[(h²+(¼)a²] √[(50)²+¼(30)²]=√(2500+225) = √2725 ≅ 52.201 in

Use your circle conjectures to solve the problem for <BAD and < BOD

Step 1 - From the circle, we have that the marked angle is <BAD = 90 Then, the angle in the center corresponding to the arc is <BOD = 180 Step 2 - So, the arc be also DAB = 180 From figure arc AD = 89 Then d = arc(AB) = 180 - 89 = 91

Pythagorean Converse Theorem

Suppose a triangle has sides of lengths a, b, and c. If a squared + b squared = c squared, then the angle opposite of side of length c is a triangle.

If a right rectangular prism has dimensions a,b, and c, then the length d of its space diagonal is found by the formula d = square root a^2+b^2+c^2.

True, √(l2 + w2 + h2) units.

a radio station is located 10 miles from a straight highway and has a signal range of 15 Mi. Approximately what length of the highway is within the station's signal range?

by the Pythagorean theorem, d=√15²-10²≅11.2 miles. does the length of the highway that is within the station's signal range is 2×diameter = 2×11.2≅22.4miles also, OS²=OP²+PS²

Find the sum of measures of its interior angles of a polygon of n sides if a) n=6 b) n=8

n side a)n=6 (n-2)180°= 4(180°)= 720° b)n=8 (n-2)180° 6(180°)= 1080°

Parametric

the <T Theta is called parameter. This is a variable that appears in a system of equations that can take on any value (unless limited explicitly) but has the same value everywhere it appears.

angle of elevation/depression

the angle formed by a horizontal line and the line of sight to an object above (below) the horizontal line.

C-64 - Two congruent chords in a circle are equally distant from

the center of the circle.

slant height

the height of a lateral side to the base of the figure in a three-dimensional figure (cone or pyramid)

C-63 - The perpendicular from the center of a circle to a chord is

the perpendicular bisector of the chord.

a dart board consists of concentric circles in line segments as shown. The radius of the bullseye is 4 inches in each ring surrounding the bullseye is 3 inches wide. what is the area of the bullseye?

πr²=4²π=16π=50.265

proof information of triangle ABC, angle bisector= CM

•the congruent legs: AC=BC • the congruent base legs: <A = <B • the altitude CM bisects the vertex < C • the altitude CM bisects the base AB

Find the equation of the line tangent to circle S centered at (1,1) if the point of tangency is

(5,4). 4=5/4x5+c =4-25/4c=-9/4 Y=4/5x-9/4 Now adjust because the center is (1,1)

Find the number of sides for a polygon whose sum of measures of its interior angles is

(n-2)×180 = 900 900/180+2= 7 ​

Trapezoid Perimeter

Add all sides

The following is a Pythagorean Identity?

sin^2x + cos^2 = 1, this identity is derived from the definitions of the sine and cosine function and the Pythagorean Theorem of Geometry.

C-113 - Pythagorean Identity

sin²θ + cos²θ = 1 For any acute angle, A, cos^2A+sin^2=1

Use the definitions of three trigonometric ratios to complete each statement.

sinѲ = s cosѲ=r tan Ѳ=s/r

if we have a circle of radius 20 with its Center at the origin, the circle can be described by the pair of the following equations:

x=20cos(t) and y=20sin(t)

what is the Circle Center is not at the origin? what equations describe its position?

x=h+rcos(t) and y=k+rsin(t)

parametric equation of a circle

x=rcos(t) and y=rsin(t)

radius of a circle

1/2 the diameter or r=√x²+y²

the diameter of the cone is 40 ft and the height is 21 ft. Find the slant height

21²+20²=s² 441+400=s² 841=s² s=√841 S≅29ft

the base of the pyramid is a square with side lengths of 30 in. The height of the pyramid is 50 inches. Find the slant height

50²+15²=s² 2500+225=s² 2725=s² s=√2725 s≅52.2in

the sides of the trapezoid are: 7, 14.32, 14.87, and 15, the altitude is 14, what is the perimeter?

51.19

a circle with the equation, x²+y²=64 has what radius...?

8

Use your circle conjectures to solve the problem for the linear pair.

<1 = 180 - 88 = 92 (linear pair) =<1 = ½(118+f) =92 = ½(118+f) = 184-118 = f = 66 = F

P. 567 - #64 - Find the area of the attached.

Area of a trapezium: ½(16+12)×8cm²= 112cm²

What is the circumference of a circle?

C = 2п(20) = 40 п is approx.. 125.6 cm

You must do things you think you cannot do.

Eleanor Roosevelt

A diameter is a line segment connecting any two points of a circle, true or false...

False, a chord is a line segment connecting any two points of a circle.

A chord is a segment connecting the center of a circle to any point of the circle, true or false...

False, a radius is a segment connecting the center of a circle to any point of the circle.

The center of a circle is a point of the circle, true or false...

False, the center of the circle is not a point on the circle.

If an isosceles right triangle, if the legs are of length x, then the hypotenuse is of length x square root 3.

False, the legs need to be of length x square root 2.

How to use the pythagorean theorem in carpentry

Given 2 straight lines the Pythagorean Theorem allows you to calculate the length of the diagonal connecting them.

Converse of the Pythagorean Formula

If the lengths of the three sides of a triangle work in the Pythagorean formula, then the triangle is a right triangle.

Converse of the Pythagorean theorem

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.

How is the pythagorean theorem used in surveying and navigation? where might the right triangle be located?

Let us consider 3 ships (stationary) ship B is on the same line of ship a and also on the same line of ship C here a right triangle is formed at ship c. let the distance between the ship a and B be 3 kilometers where Bc is 4 km AC²=BC²+AB² AC²=3²+4²=25=7 AC= √25 = 5km

The angle of elevation from a sailboat to the top of a 121 - ft lighthouse on the shore measures 16 degrees. To the nearest foot, how far is the sailboat from shore?

Tan 16 degrees = 121/d d(tan 16 degrees) = 121 D = 121/tan 16 degrees The sailboat is approximately 422 feet from shore.

draw and label radius at base of cone, what do you get?

a right triangle is formed when the radius and height are are legs and the slant is the hypotenuse

Biconditional

a statement that contains the words "if and only if"

sides of equilateral triangle

a=b=c

Counterexample

an example used to support a claim or statement that is the opposite of another claim or statement

angles of an equilateral triangle

are equal - each is 60, A=B=C=60°

Pythagorean Formula

a²+b²=c²

semicircle are constructed on each side of triangle ABC shown above angle C is a right triangle with a c equal to 6 units and BC with 8 units. What is the sum of the areas of the semi-circles

by the Pythagorean theorem Abbie question units does the sum of the areas of the semi-circles is given by ½π(3)²+½π(4)²+½π(5)²=25π

acute triangle

c²<a² + b²

right triangle

c²=a²+b²

obtuse triangle

c²>a² + b²

Pythagorean Theorem

for the right triangle, the side that is opposite the right angle is called a hypotenuse. the other 2 shorter sides are called legs. The pythagorean theorem states that if you add the squares of the lengths of the two legs of a right triangle, you will always obtain the square of the length of the hypotenuse.

altitude of equilateral triangle formula

h=½×√3×a

the radius of the cone is 14 cm and the slant height of 28 cm. Find the height of the cone

h²+14²=28² h²+196=784 -196=-196 h²=√588 h≅24.2cm

in a pyramid or cone the height will always be

perpendicular to the base meaning it intersects the base at a 90 degree angle

C-61 - If two chords in a circle are congruent,

then they determine two central angles that congruent.

Lindsay wants to mail a candle that is 8 in tall to her friend for her birthday. She's at the store looking for a box to ship the candle in. Lindsay needs to know if the candle will fit diagonally in the Box? the bottom base long side is 4 in, height of box is 7in, and short side 3in?

yes 1. The diagonal distance of the base: 4²+3²= c², c=5in 2. The diagonal of the 3D triangle is: 5²+7²=c², c=√74≅8.6in

how to apply the Pythagorean Theorem to the 3 dimensional figure (attached)...

you can fo this by adding a triangle to the cube. 5²+5²=x², x= 7.1

solve for the coordinates in a circle using what formula?

y±√r²-x²

draw and label radius at base of cone, what do you get?

a right triangle is formed when the radius and height are the legs and the slant height is the hypotenuse

Given: isosceles trapezoid, BEAR, find AQ, with large BEAR length= 27, and small length= 15...steps...?

1. Figure out lengths of top and bottom of trapezoid. 2. The triangles on both ends, are congruent according to HL. 3. The missing side can be found using the pythagorean theorem. * remember to decompose the two right triangles and rectangle to solve.

how does a circle work with the Pythagorean theorem?

1. In a right triangle, we can see sinϴ=y/r, cosϴ=x/r; recall the trig identity sin²ϴ+cos²ϴ=1 2. Substitute x/r and y/r into the identity, (y/r)²+(x/r)²=1 3. Remove the parentheses, y²/r²+x²/r²=1 4. Multiply through by r², y²+x²=r²

Consider a quadrilateral which to each of the diagonal measures 27 cm. The length of the adjacent sides are 17 cm and 21 cm. Determine whether the quadrilateral is a rectangle.

1. Length of diagonal is 27cm. Hence AC=BD= 27cm. 2. The length of the adjacent sides is 17cm and 21cm, AD= 17cm, AB=21cm 3. Consider the. ⃤ ABC, now find whether ABC is a right ⃤or not AC²? AB²+ BC² 729≠731 ABC is not a right. ⃤ and so B≠90°. Again, in order to be a rectangle all <'s must be 90°. This does not happen. ∴ the quadrilateral is not a rectangle.

Triangles ABC and DEF are similar. Find the perimeter of triangle ABC.

1. Take the properties of △ABC BC/EF= AB/DE (6/9)= AC/DF 2. ⅔(4) [since DF=12]=8/12 3. ∴AC=8 (10+6+8)in = 24in

find the length of diagonal AB if the long base length of line segment AC is 12 CM, the short base length of CD is 5 cm and the height of line segment BD is 2 centimeters.

1. The diagonal distance of the base: 12²+5²= c², c=13cm 2. The diagonal of the 3D triangle is: 13²+2²=c², c=√173≅13.2cm

. It looks like the moat contractor doesn't quite understand the concept of a moat. If the circular moat should have been a circle of radius 10 meters instead of radius 6 meters, how much greater should the larger moat's circumference have been?

2∏r = 20∏, 12∏, the circumference should have been the difference of 20 and 12∏

the sides of the trapezoid are: 11, 8.94, 17, and 8.25, the altitude is 8, what is the perimeter?

45.19

the sides of the trapezoid are: 7, 13, 15, and 12.37, the altitude is 12, what is the perimeter?

47.37

a window washer is repelling and cleaning the windows of a hotel shaped like square pyramid shown. The height of the pyramid is 200 feet. If the window washer travels to the top of the pyramid and back down along a straight path, X, how many total feet has he traveled?

500ft=increase 250ft+decrease 250ft 200²+150²=x² 40000+22500=x² 62500=x² s=√62500 X=250ft

A circle is the set of all points in space at a given distance from a given point, true or false.

False, a sphere is the set of all points in space at a given distance from a given point.

A tangent is a line in the plane of a circle, containing two points of the circle, true or false...

False, a tangent is a line in the plane of a circle, containing one point of a circle.

In two different circles, arcs with the same measure are congruent, true or false...

False, in two different circles, arcs with the same measure are congruent only If the circles are congruent.

In a 30-60 right triangle, if the shorter leg is of length x, then the longer leg is of length 2x and the hypotenuse is of length x square root 3.

False, the hypotenuse length is 2x and the longer leg length is x square root 3.

In an isosceles right triangle, if the legs are of length x, then the hypotenuse is of length x square root 3.

False, the legs need to be of length x square root 2.

The measure of an arc is equal to one half the measure of its central angle, true or false...

False, the measure of an arc is equal to the measure of its central angle.

Two circles are congruent if they have the same center, true or false...

False, two circles are concentric if they have the same center.

C-114 - Law of Cosines

For any triangle with sides of lengths a,b, and c, and with C the angle opposite the side with length c, c^2 = a^2+b^2-2abcosC

An interior angle of a parallelogram has a measure of 35ﾟ. What are the measures of the other 3 interior angles in a clockwise order?

Let ABCD be the parallelogram and let the measure of m> a equal 35ﾟ. A parallelogram is a quadrilateral with 2 pairs of parallel sides. The opposite or facing sides of the parallelogram are of equal length and the opposite or facing angles of a parallelogram are of equal measure. Here <A, <C, <B and <D are opposite angles of the parallelogram. Since opposite sides of a parallelogram are all equal in measure and measure of angle a is 35. The measure of angle C is also 35. 2. now use the fact that the sum of the measures of the interior angles of a parallelogram = 360ﾟ. In other words if ABCD is a parallelogram then m<A+m<B+m<C+m<D=360° 3. now substitute 35ﾟ for the measure of angle a and measure angle C and simplify further to solve for angle B and angle D.. 35°+m<B+35°+m<D=360° 70°+m<B+m<D=360° Add m<B+m<D = 290° subtract 70° from both sides of the equation 4. Since m<B and m<D are opposite angles of the parallelogram, both angles are equal in measure which is 290ﾟ / 2 or 145ﾟ hence m<A= 35°, m<B= 145°, m<C= 35ﾟand m< D equals 145ﾟ. thus the measure of the other 3 interior angles in a clockwise order are: 145°, 35°, 145°.

Find the length of chord CE

Let M be the midpoint where line BD and CE will intersect each other Now BD = 20in DM= 5in BD= 25 in So the radius of the circle is 25". Also BC is the radius of the circle so BC is 25"

How is the pythagorean theorem used in surveying and navigation? where might the right triangle be located?

Let us consider 3 ships (stationary) ship B is on the same line of ship a and also on the same line of ship C here a right triangle is formed at ship b. let the distance between the ship a and B be 3 kilometers where Bc is 4 km AC²=BC²+AB² AC²=3²+4²=25 AC= √25 = 5km

Use your circle conjectures to solve the problem for <AOB

The angle between the radius and tangent line to a circle is a right angle. Step 1: Find the angle subtended at center in the right triangle AOB <A+<B+<O = 180 35+90+<O = 180 <O = 55 So, the length b of the given arc is given by the formula L = rѲxп/180 B= rx55x п/180 = .96r

A circle with center L contains points J and K. Circle L is dilated by a scale factor of 2, resulting in a new circle with center P. Points M and N are on the circle P such that the central angle MPN has the same measure as central angle JLK. What can we say?

The arc length of J k's half the arc length of MN

A chord is a segment connecting the center of a circle to any point of the circle, true or false...

false, a radius is a segment connecting the center of a circle to any point of the circle.

C-62 - If two chords are congruent,

then their intercepted arcs are congruent.

C-61 - If two chords in a circle are congruent,

then they determine two central angles that are congruent.

C-65 - The perpendicular bisector of a chord passes

through the center of the circle.

The center of a circle is a point of the circle, true or false...

true, the center of the circle is the midpoint of the diameter

basic equation of a circle

x²+y²=r², where X and Y are the coordinates in radius ( hypotenuse) of the circle the result is a Pythagorean theorem