Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.
Two right triangles C D A and C D B share a common side C D. Side A C equals 2 x, and side B C equals 3 x minus 5. Right angle is at D.
In the figure,
is the perpendicular bisector of
.
If the length of
is 2x and the length of
is 3x − 5, the value of x is
.
The value of x is 5.
To find the value of x in the given figure, we need to equate the lengths of the sides of the triangle.
Given: AC = 2x and BC = 3x − 5
Since CD is the common side of both triangles, we can equate the two sides:
AC = BC
2x = 3x − 5
Now, we can solve for x:
2x − 3x = −5
−x = −5
x = (-5)/(-1)
Therefore, the value of x is 5.
Answer: 5
To find the value of x, we need to equate the lengths of the sides of the right triangles.
According to the given information, side AC is equal to 2x and side BC is equal to 3x - 5.
Since AD is the perpendicular bisector of BC, it divides BC into two equal parts, so BD is also equal to (3x - 5).
In right triangle ADC, using the Pythagorean theorem, we have:
AC^2 = AD^2 + DC^2
(2x)^2 = (3x - 5)^2 + DC^2
4x^2 = 9x^2 - 30x + 25 + DC^2
0 = 5x^2 - 30x + 25 + DC^2
We can simplify this equation further by substituting DC with BD:
0 = 5x^2 - 30x + 25 + (3x - 5)^2
Simplifying further:
0 = 5x^2 - 30x + 25 + 9x^2 - 30x + 25
0 = 14x^2 - 60x + 50
Now, we can solve this quadratic equation by factoring or using the quadratic formula. However, since we are asked to type the answer in the box using numerals, we can proceed using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation:
a = 14, b = -60, and c = 50.
x = (-(-60) ± √((-60)^2 - 4 * 14 * 50)) / (2 * 14)
x = (60 ± √(3600 - 2800)) / 28
x = (60 ± √800) / 28
x = (60 ± 20√2) / 28
Now, for the answer to be in numerals, we need to simplify this further:
x ≈ (60 + 20√2) / 28
x ≈ (15 + 5√2) / 7
Therefore, the value of x is approximately (15 + 5√2) / 7.