Use the following image for part (b)
To prove the first part of (b), note that the medians intersect at a point 2/3 of the distance from their respective vertexes.
And let this point be G.
By the triangle inequality, we have that
AC < (2/3)AD + (2/3)CF
BC < (2/3)BE + (2/3)CF
BA < (2/3)BE + (2/3)AD
Adding these, we have that
AC + BC + BA < (4/3) [ AD + BE + CF ]
And AC + BC + BA = the perimeter, P, of triangle ABC
And AD + BE + CF = the sum of the median lengths
Multiplying both sides by (3/4) we have that
(3/4) P < AD + BE + CF (1)
To prove the second part of the inequality.....construct circles c, d and e such that c has radius AD, d has radius, CF and e has radius BE
Then because AD cuts AC at H and CF cuts BC at I and BE cuts AB at J, then we have that
AD < AC
CF < BC
BE < AB thus
[AD + CF + BE] < AC + BC + AB which implies that
[ AD + BE + CF ] < P (2)
And......putting together (1) and (2)
(3/4)P < [ AD + BE + CF ] < P
Here's another method....
The sides are all equal, so one side = 148/ 4 = 37
And the diagonals meet at right angles......so 1/2 the length of the known diagonal = 1/2 * 24 = 12
And this half diagonal and one of the rhombus sides will form the leg and hypotenuse of a right triangle.....and the other half diagonal will form the other leg.....and its length = sqrt [ 37^2 - 12^2] = sqrt [ 1225 ] = 35
So......the length of the other diagonal = 2 * 35 = 70
The sum of the digits of a two digit number is 9. the fractions formed by taking 9 less than the no.. as numerator and 9 more than the number as denominator is 3/4. what is the number
Let the digits be a and b
Note that we can write any two-digit number as 10a + b
So we have this system
a + b = 9
[ (10a + b) - 9] / [ (10a + b) + 9] = 3/4
Manipulating the first equation, we have that b = 9 - a
Subbing this into the second equation, we have that
[ (10a + 9 - a) - 9] / [ (10a + 9 - a) + 9 ] = 3/4 simplify
[ 9a] / [ 9a + 18] = 3/4 cross-multiply
4[9a] = 3 [9a + 18 ] simplify
36a = 27a + 54 subtract 27a from both sides
9a = 54 divide both sides by 9
6 = a
And b = 9 - a = 9 - 6 = 3
So....our number is 63
Proof that this is correct :
[63 - 9] / [63 + 9 ] = 54 / 72 = [3 * 18 ] / [ 4 * 18 ] = 3 / 4
Factoring and tell whether the parabola opens up or down. Also Identify the minimum and how does the axis of symmetry relate to the x-intercepts?
Note that the form is y = x^2 + bx + c
factoring and tell whether the parabola opens up or down. Also Identify the minimum and how does the axis of symmetry relate to the x-intercepts?
Since x^2 is positive and there is no negative in front of the y, term, this parabola opens upward
Factoring we have.... y = ( x - 8) ( x + 4)
Setting y = 0, we can find the x intercepts thusly : 0 = (x - 8) ( x + 4).....setting each of these factors to 0 and solving for x produces the x intercepts of x = 8 and x = -4
The minimum can be found thusly :
The x coordinate of the vertex is given by -b / [ 2a ] ....b = -4 and a = 1 .....so .... -b / [2a ] = -[ -4] / [ 2(1)] =
Now....we can find the y coordinate of the vertex by plugging this value into the function...so we have ...
y = (2)^2 - 4(2) - 32 = 4 - 8 - 32 = 4 - 40 = -36 .....and this is the minimum y value of the parabola
The axis of symmetry will be found between the intercepts and is given by adding the intercepts and dividing by 2....so we have ... [ 8 + -4 ] / 2 = 4 / 2 = 2.......so the axis of symmetry is x = 2....this is no coincidence.....the axis of symmetry will occur at the vertex which is ( 2 , - 36)
Kola and Bola are two friends. They are staying on two sides of a pond. Now Kola jumped 1 inch first to reach Bola. Seeing that, Bola jumped double distance of Kola. This time, Kola jumped equal distance of Bola. Thus they could meet each other by jumping equal distance. If the distance of two sides of the pond is 1533 inches, then how many jumps they have given totally?
With the first three jumps, they have closed a distance of 1 + 2 + 2 = 5 inches
So...there are 1533 - 5 = 1528 inches left between them....if each jumps 2 inches at a time.....they will each jump a total of
1528 / 2 = 764 inches......so....it will take each 764 / 2 = 382 jumps to meet each other [ assuming that they can jump in water !!! ]
Note that in the 12 hours after 12 noon, the clock hands pass each other 11 times until they meet again at 12 midnight
So....that means that they pass each other every 12 / 11 ths hours...
So....in terms of minutes ...... 12/11 * 60 ≈ 65.45 minutes = 65 minutes and [ .45 *60 ] seconds ≈
65 minutes and 27 seconds
So....they meet again at about 27 seconds after 1:05 PM