Maths Problem (HELP ME AND GET 10 POINTS)?

Question

The length of the sides forming right angle of a right angled are 5x and (3x-1)cm.If the area of the triangle is 60cm sq,find its hypontuse.

Answer 1

Ok, you've got a good one here.
For a right triangle, height and base are the short sides; and area of the triange is (bh)/2. Since we know the area, and the length of the sides, we can plug it in to an equation, and find x. So...
A = (bh)/2
60 = ((5x)(3x-1))/2
120 = (5x)(3x-1)
120 = 15x^2 - 5x
15x^2 - 5x -120 = 0
Divide both sides by 5 to get
3x^2 - x -24 = 0
Ok, there's a number of ways to do this, but the simplest way is to list the combinations of integers that will multiply to get -24 (the constant in the above equation). That gives us the combinations of (1, -24; 2, -12; 3, -8; 4, -6; 6, -4; 8, -3; 12, -2; and 24, -1). Now, we need to figure out the one combination that, when the second number is multiplied by three (the coefficient of x^2), you can add it to the first number to become (-1). It happens to be (8, -3) {-3*3=-9, -9+8 = -1}. There; you've factored the equation, leaving you with...
(3x+8)(x-3)=0
Since multiplication rules say that when ab=0, either a=0 or b=0, we can test by setting both numbers equal to zero; giving us...
3x+8 = 0 ; x-3 = 0
x = -8/3; x = 3
Since x is an absolute length, and lengths can't be negative, we can disregard (-8/3) for now. That gives us x=3, and putting that back into the original lengths gives us sides of 15 and 8. To find the hypotenuse, we use Pythagorean Theorum: a^2 + b^2 = c^2, substituting in the known lengths gives us:
15^2 + 8^2 = c^2
225 + 64 = 289 = c^2
c = √289 = 17
So, the hypotenuse of your triangle is 17.

Answer 2

Since the sides of the triange are 5x and (3x-1)
and you know that the area is 60

you can set up the following expression:

(1/2)(5x)(3x-1)=60

solving for x...you get 3

once you have x...you can figure out the lengths of the sides

side A=5x=(5)(3)=15
side B=3x-1=(3)(3)-1=9-1=8

and with the length of the sides, you can use the pythogorean thorem

A^2 + B^2 = C^2

plug in the values for the sides for A and B respectively

(15)^2 + 8^2=C^2
C^2=289

take the square root of the answer

C=sqrt(289)

and you get plus/minus the length of the hyotnuse

C=+17,-17

throw out the negative b/c we're looking for a distane

the hypotnuse=17

Answer 3

The area of a triangle is 1/2 base x height.

so let 5x be the base and (3x-1) the height

so 1/2 5x (3x-1)= 60

multiply everything

7.5x^2 + 2.5x = 60

get everything to one side

7.5x^2 + 2.5x - 60 = 0

now solve using the quadratic formula.

x = -3 or 8/3

since a side of a triangle can't be negative x = 8/3

so the base = 40/3 and the height = 7

now use Pythagorean theorem

A^2 + B^2 = C^2

so the hyp is 15.1

Answer 4

the srae of a right triangle is 1/2 x (base) x (height)
here the height is the perpendicular so let it be 5x and base be 3x-1 just put in the values and ull get the answer and also area is also give 60 so ull get x.
60=1/2 x (5x) ( 3x-1)
120= 15x(square) - 5x
ull get 15x(square) -5x -120=0
3x(square)-x - 24=0
3x(square)-9x+8x-24=0
3x(x-3)+ 8(x-3)=0
(3x+8)(x-3)=0
x=-8/3 or x= 3
neglecting x=-8/3
now the two sides are 5x= 5 X 3 =15
and 3x-1= 3X 3 -1=8
by pythagorous theoram p(sq.) + base(sq.)= hypotenuse(sq.)
225+ 64=h(sq.)
h=17

Answer 5

there are two answers to this qns.

the hypotenuse should be 17.0cm (to 3sig. fig) or 16.1cm (to 3sig. fig) depending on the x-value that's chosen. in this case, one should note that both x-values are mathematically sound.

Total area of right-angle triangle = 1/2 x base x ht

therefore to find x:
(3x - 1)(5x)(1/2) = 60
solve and you'll reach 3x^2 - x - 24 = 0

x = 3 or x = -2 2/3

both values are accepted. we do not reject the negative x-value because further on, a check using Pythogoras' Theorem will show that the hypoteneus is also positive.

Let C be the unknown length of the hypoteneus

Method 1:
Using Pythagoras' Theorem using x=3,
C^2 = (5x)^2 + (3x - 1)^2
C^2 = 289
C = (289)^(1/2)
thus, C=17cm (to 3sf)

Method 2:
Using Pythagoras' Theorem using x = -2 2/3,
C^2 = (5x)^2 + (3x - 1)^2
C^2 = 258 7/9
C = (258 7/9)^(1/2)
thus, C= 16.086 = 16.1cm (to 3sf)

Answer 6

Because two of these right triangles can form a rectangle, so the angle between the hypotenuse and the adjacent side of the triangle is 45 degrees.
Then use tangent:
tan45degrees=adjacent side/ opposite side
But first, remember to find x: (5x)(3x-1)/2=60

Answer 7

you're going approximately this subject the incorrect way. First set the equation equivalent to 0. 0= -X^2+2x+4 (observe the neg. image in front of the X^2. This equation can't be factored or "disbursed into paretheses" as you call it so which you could use the quadratic formulation. the quadratic formulation is composed of three variables... A, B, and C. For this equation you have A= -a million, B=2, C=4. Plug in and remedy for X. X= a million+2?5 /-2 simplify and you have your answer! for sure no matter what you do you will no longer get a ?6 so C is out... and if A is right then what took place to the two formerly the ?5? So A is misguided additionally... that leaves you with... B. very final answer.

Answer 8

First of all I would like you to understand that such problems should be solved by yourself...... just because you have these facilities dont make it a way to do your homework ..may be this was too tuff for you but I request you to look into the problem yourself till you donot find an answer Anyways your solution is:

First solve this equqtion for x:
60= 1/2 X 5x X (3x-1)
[i.e. area=1/2 X height X length]
solving this quardratic equatin you will get x=3
therfore length of the 2 sides are 5x=15 and 3x-1=8.
Now using pythagoras theorm sove for length of hypotaneous:
therefore answer=17cm

Answer 9

17

Answer 10

(1/2)*(5x)(3x-1) = 60
5x*(3x-1) = 120
15x^2 - 5x = 120
15x^2 - 5x - 120 = 0
3x^2 - x - 24 = 0
x = (1+sqrt(1-4*3*(-24))/(2*3) = 3

Applying the Pythagorean theorem:
(5*3)^2 + (3*3-1)^2 = h^2 = 289
h = 17