Geometrical Problem! Look below for the picture & question.?

Question

In the adjacent squares shown, the vertices A,B, and C lie in a straight line. The value of x is: (what is the exact value?) Also, explain your answer briefly as well, thanks.


http://www.e-zgeometry.com/pow/pictures/24.gif

Answer 1

First, we need to find the relationship between the squares and triangles. You can make 2 triangles by connecting A, B, and C. Since we know that the line is straight, the triangles must be similar.

Therefore,
4/7=(7-4)/(x-7)

All four sides of a square are congruent, so to find the length of the short leg of each right triangle, we must subtract the length of the smaller side of the square.

Now that we have our equation, all that we must do is solve for x.

Cross multiply.
4(x-7)=7(7-4)
4x-28=21
4x=49
x=12.25

I hope that this helps.

Answer 2

We're talking about a line here, so start with a simple linear equation.

y = mx + b

x in this equation does not represent x in the picture - x is just the usual letter used for this form of a linear equation.

Here y is the vertical height of a vertex, m is the slope of the line connecting the vertices, x is the horizontal position of a vertex, and b is the height of the vertex when x = 0.

At vertex A, we can assume that x = 0. Thus, b = (the height of vertex A) = 4.

At vertex B, we have y = (the height of B) = 7, and x = (horizontal position of B) = 4, so putting these values into the linear equation lets us solve for m:
7 = m (4) + 4
implying that m = (7-4) / 4 = 3 / 4
So our equation to find the height of a vertex is
y = 3 x / 4 + 4

Checking this equation with vertex A correctly gives y = 3 (0) / 4 + 4 = 4.
Checking with vertex B correctly gives y = 3 (4) / 4 + 4 = 7
Finding the height of C gives y = 3 (11) / 4 + 4 = 33 / 4 + 4 = 8.25 + 4 = 12.25

The height of C is the same as the width of the largest square, so the exact value for x in the picture is 12.25

Answer 3

If you draw a straight line connecting all 3 points, you will form 2 right triangles. The first one is a standard 3-4-5 right triangle (4 from the side of the square, 3 from the projecting edge of the 7 square (7-4=3), 5 hypotenuse). Using the cosine function, we find that the interior angle formed by the hypotenuse and the 4-side is 36.8698 degrees.

Using the law of equal interior angles, we know that the interior angle of the triangle bounded on the bottom by the 7-square is also 36.8698 degrees. Using the tangent function with the one known side and the interior angle, we find that the short segment of the largest triangle is 5.2499 (5.25 rounded). Adding 7 for the longer segment of the "B" square gives us a value of 12.25 for x.

Answer 4

First, let's use z instead of x to avoid any confusion with Cartesian coordinates.

Picture this drawing in a Cartesian plane with A being the origin, (0,0). Then B=(4,3) and C=(11,z-4).

The line joining A and B is then y = 3x/4. Plug x=11 for point C and you get
z-4 = y = 3*11/4 = 33/4
→ z = 33/4 + 4
→ z = 49/4.

Answer 5

The line between A and B has a slope of 3/4

Therfore, the vertical distance between C and B must be equal to 7 * 3/4 = 5.25

Thus, x = 7 + 5.25 = 12.25

Answer 6

Well if A, B, and C is in a straight line, then the rate of increase is constant. There is an increase of 3/4 between A (1,4) and B (4,7). Then there would be an increase of y/7 to get to C. Using proportions I get y=5.25. Add that to 7 and you get 12.25.

I hope this helps

Answer 7

3/4 = (x - 7)/7
4x - 28 = 21
x = 49/4 = 12.25

Answer 8

similar triangles:
3/4 = (x-7)/7
21 = 4x - 28
4x = 49
x = 12.25

Answer 9

? = distance of square with side 7 to the top of square with side x

The rise over run of a to b is 3:4
The rise over run of b to c is ?:7

So 3/4 = ?/7
21 = 4?
21/4 = ?

21/4 + 7 = x
21/4 + 28/4 = x

49/4 = x
or
12 1/4 = x
or
12.25 = x