CPhill

Let the bottom left vertex of the right triangle lie at (0,0).....so.....the square will touch two sides of the triangle

Let the height of the right triangle = 5 and the base length = 12

And the endpoints of the hypotenuse lie at (0,12) and (5,0)

And the equation of the line conecting these two points is given by :

y = -(12/5)x + 12  (1)

And the diagonal of the square will be the segment lying on the line y = x    (2)

Find the x coordinate  intersection of  (1) and (2)  ..... this will be the length of the side of the square

So we have

- (12/5)x + 12  =  x      add -(12/5) to both sides

12 =  (17/ 5) x       multiply both sides by  5/17

60/17  units =   the side of the square

-6x+5y = 1

6x+4y = -10

Just add these and we get

9y = -9   divide both sides by   9

y  = -1

Sub this into either equation to find x

6x + 4(-1) = - 10

6x - 4 = -10      add 4 to both sides

6x = - 6     divide both sides by 6

x = - 1

Nope    ...2  x 6  =   12

We don't have "7" on a die....

So....which other one pairs a letter and a number ??

DCO....since CO = DO.....the angles opposite those sides are equal

So DCO = x

CBO....angle OBQ = 90°  [tangent meeting a radius forms a right angle ]

So CBO = 90 - 2x

OCB   .......since OB = OC, the angles opposite those sides are equal

So OCB = CBO =  90 - 2x

BCD = DCO + OCB =  (x) + ( 90 - 2x) =  ( 90 - x)   [angle addition postulate ]

DOB is a central angle that is twice the measure of BCD = 2(90 - x) = 180 - 2x  = measure of minor arc   DB   [ a central angle measures twice  an inscribed angle intercepting the same arc ]

Note that arc DCB = 360° - minor arc DB  =  360 - (180 - 2x) = 180 + 2x

[ the sum of arcs in acircle = 360° ]

Angle  y  =  (1/2) [ difference of  arc DCB - arc DOB] 

[ The angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the difference of the intercepted arcs ]

Read more: https://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php#ixzz5YI50pVkl

(1/2) [ (180 + 2x) - (180 - 2x) ]  =

(1/2) [  2x + 2x ]  =

(1/2) [ 4x]  =   

2x

1a. What would be the greatest height the telescope could reach above the ground?

The greatest height will be when a = 0  =  7 meters

1b. What is the diameter of the telescope?

We can find the a value where the hight of the telescope = 7 m on the right side 

7 = a^2 -  a + 7     subtract 7 from both sides

0 = a^2 - a

a^2 - a  =  0     factor

a ( a - 1)  =  0

Setting both factors to 0 and solving for a  gives that

a =  0     (we know this)     or  a =1

So.....the diameter of the telescope is 1 meter

1c. What is the depth of the telescope?

The low point of the telescope will be where   a = (1/2) 

So....to find the height at that point, we have

(1/2)^2 - (1/2) + 7  =  1/4 - 1/2 + 7  =  7 - 1/4  =  6.75 m

So.....the depth of the telescope is    7 -  6.75  =  .25m  

Since BC bisects angle ABQ then angle ABC = angle CBQ = x

[Definition of an angle bisector ]

Since angle CBQ = x, then minor arc  BC = 2x

[ A tangent meeting a chord forms an angle that is 1/2 the measure of the arc intercepted by the chord ]

Angle A =  (1/2 ) minor arc BC  =  (1/2)(2x)  = x

[ A is an inscribed angle....ts measure = 1/2 of its intercepted arc ]

Since angle A = angle ABC, the sides opposite these angles are also equal

So AC = BC

[ In a triangle.....sides opposite equal angles are also equal ]

Minor Arc  AB  = (2) 72  = 144°

[  A tangent  meeting a chord forms an angle half that  of the arc intercepted by the chord ]

Angle C  =  (1/2) measure of minor arc AB = (1/2) 144 = 72°

[ C is an inscribed angle.....its measure is 1/2 the measure of its intercepted arc

Angle A =  180 - 72 - 54  =  54°

[ The sum of the interior angles of a triangle = 180° ]

So.....triangle ABC is isosceles because angles A and B are equal

Example .....to find a side length.....use the distance formula

Are you familiar with this???

Distance from A to B   =

sqrt  [ ( 1 - 2)^2 + (3 - 0)^2 ]  = sqrt [ (-1)^2 + 3^2] = sqrt [ 1 + 9 ] =  sqrt (10)

Then find the distances between B,C     C,D      and      D,A

They should all be =  sqrt (10)

Let me know if you have trouble

2 outcomes for flipping the coin and 6 outcomes for rolling the die

So....the number of total  outcomes =  

No. of Outcomes of first thing  x  No. of Outcomes of second thing

Based on this.....what is the answer, NSS???