CPhill

1. Segment line AD bisects Angle BAC. If BD=DA-AC, what is Angle C in degrees?

I think you must mean that  BD = DA = AC    

[ DA - AC   couldn't be as long as BD ]

Since BD = AD, then angles BAD and  ABD are equal

And angle CAD = angle BAD

And since DA = AC, then angle  ACD = angle ADC

Let the measures of angles ABD, BAD, CAD = x

And let  the measures of angles ACD and ADC = y

So we have that

x + x + x + y = 180   ⇒    3x + y = 180   ⇒ y = 180 -3x       (1)

x + y + y = 180  ⇒    x + 2y = 180        (2)

Sub (1) into (2)

x + 2(180 - 3x) = 180

x + 360 - 6x   = 180

-5x =   - 180

x = 36°

And angle ACD = angle C =   180 - 3(36) = 180 - 108   =  72°

Here's my take on this

P ( Red l Red )   =    P ( Red and Red )

                                _______________

                                     P (Red)

We can disregard the three Black / Black cards  [ since we already  hold a Red ]

Of the three remaining cards, we had (2/3) of a chance of choosing a Red/Red card

And of the three remaining cards, we had a 5/6 chance of  choosing a "Red" side showing first

So......the probability is

(2/3) / (5/6)   =   (2/3) (6/5)  = 12 / 15   =   4 / 5

5. Point H is the orthocenter of Triangle ABC. If Angle BAC=115 degrees, what is angle BHC in degrees?

angle BAC = 115....so angle BAF = 180 - 115 = 65°

And angle AFB = 90

So angle ABF = 180 - 65 - 90 = 25°

And in triangle  BHE, angle BEH = 90°

So....angle BHE = angle BHC = 180 - 90 - 25 = 65°

4. Point O is the circumcenter of Triangle ABC. The distance between O and Line AC is 7, and the distance between O and Line AB is 15. If AC=48, what is AB?

OD is a perendicular bisector of AC

So.....DA = (1/2)AC = 24

And triangle ODA is a 7 - 24 - 25  Pythagorean Right Triangle

So OA = 25

And OE = 15

So....since OE is perpendicular to AB....triangle AOE is a right triangle

And   EA = √ [ OA^2 - OE^2 ] = √ [ 25^2 - 15^2] = √ [ 400]   = 20

And since OE is a perpendicular isector to AB, then AB = 2EA = 2 * 20  = 40

3. In the diagram below, O is the circumcenter of Triangle ABC. If Angle BAC=28 degrees and Angle OAC=32 degrees, then what is the degree measure of Angle AOC?

OA and OC will be the radii of the circumcircle.....so....they are equal

So....in triangle AOC.....the angles opposite OA and OC will also be equal

So....angles AOC and OCA are equal = 32°

So.....angle AOC   =  180  - 2(32)   =  180 - 64   = 116°  

2. Triangle ABC is an isosceles right triangle where Angle A=90 degrees. O is the circumcenter of Triangle ABC. What is Angle AOB in degrees?

In a right triangle.....the circumcenter is at the midpoint of the hypotenuse

So.....OB = OC

Since ABC is isosceles, Angle ABC = Angle ACB =  45°

And AC = AB

And AO = AO

And by SAS, triangle AOC is cogruent to triangle AOB

But angle CAO = angle BAO  ....so AO bisects angle BAC

So angle BAO = 45°

And angle ABO = angle ABC = 45°

So....in triangle AOB....angle AOB = 90°

1. Triangle ABC has circumcenter O. If AB=10 and Area of OAB=30 find the circumradius of triangle ABC

AB is a chord of the circumcircle

And we can find the altitude of triangle oAB thusly

30  =  (1/2)AB * altitude       multiply through by 2

60 = 10 * altitude     divide both sides by 10

6 = the altitude

But.... because the altitude is perpendicular to AB, the altitude will bisect chord AB

Call the bisection point, M

So....we have right triangle   MOA

And OA  is  the hypotenuse of this triangle = the circumradius

And AM is one leg = 6

And MO is the other leg = 5

So.....using the Pythagorean Theorem

OA = √ [ AM^2 + MO^2] = √ [ 6^2 + 5^2 ] = √ [ 61 ]  = the circumradius

Every real number  in the interval 2 < x ≤ 5  will be closer to 4 than to 0

So   3/5   =   .6

y(x) = x(x + 3)   =  x^2 + 3x

(a)  We have this difference quotient 

[ (x + h)^2 + 3(x +h) - (  x^2 + 3x) ]

_________________________               simplify

            h

[  (x^2 + 2xh + h^2 +  3x + 3h)   - (x^2 + 3x) ]

___________________________________

                   h

[ 2xh + h^2 + 3h - ]

________________

           h

h [ 2x + h + 3 ]

____________

        h

2x + h + 3          and when h → 0, we have

2x + 3    ...... this is the gradient at any x  value on the curve

(b)   The gradient  of the curve at (-5, 10) is

2(-5) + 3   =      -7

Here's my best attempt, ME....

Probability  of rollling

2 on first pair and 2 on second pair   =   (1/36) * (1/36) = (1/36)^2

3 on first pair and 3 on second pair  =  (2/36) * (2/36) = (2/36)^2

4 on first pair and 4 on second pair = (3/36) * (3/36) = (3/36)^2

5  and 5      =    (4/36) * (4/36)    =   (4/36)^2

6 and 6      =     (5/ 36) * ( 5/36)   = (5/35)^2

7 and 7       =    (6/36) * (6/36)

8 and 8  = (5/36)^2

9 and 9  = (4/36)^2

10 and 10 = (3/36)^2

11 and 11 = (2/36)^2

12 and 12 = (1/36)^2

So.....the probability is

( 2 [ 1^2 + 2^2 + 3^2 + 4^2 + 5^2] + 6^2 )  /  6^4     =   

146 / 1296  = 

73 / 648