If CFD=12a+45, find a so that FC is perpendicular to FD
12a+45=90
-45=90
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12a=45
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12 12
a=45\12
To find the value of "a" such that line FC is perpendicular to line FD, we need to consider their gradients (slopes) and use their relationship.
The gradient of line FC can be determined by comparing the coefficients of "a" in the given equation, CFD=12a+45. Since there is no "a" term in the equation, it implies that the gradient of line FC is zero. In other words, FC is a horizontal line.
On the other hand, the gradient of line FD can be determined from the equation CFD=12a+45, where the coefficient of "a" is 12. Thus, the slope of FD is 12.
For two lines to be perpendicular, the product of their slopes must be -1. So, we can set up the equation:
Slope of FC * Slope of FD = -1
0 * 12 = -1
0 ≠ -1
This equation has no solution since the product of the slopes is not -1. Therefore, there is no value of "a" where line FC is perpendicular to line FD.
To determine the value of "a" that makes the vectors FC and FD perpendicular, we need to find their dot product and set it equal to zero. The dot product of two vectors is zero if and only if they are perpendicular.
First, let's find the vectors FC and FD. Given CFD = 12a + 45, we can rewrite it as:
FC = C - F
FD = D - F
Now, let's represent vector C as (x₁, y₁) and vector D as (x₂, y₂). Vector F will remain as (45, 0) since it is given.
FC = (x₁ - 45, y₁ - 0) = (x₁ - 45, y₁)
FD = (x₂ - 45, y₂ - 0) = (x₂ - 45, y₂)
Next, we'll calculate the dot product of FC and FD:
FC · FD = (x₁ - 45)(x₂ - 45) + y₁y₂
Since we want FC and FD to be perpendicular, the dot product should equal zero:
(x₁ - 45)(x₂ - 45) + y₁y₂ = 0
Now, we can substitute CFD = 12a + 45 into the equation:
(12a)(a - 45) + (12a + 45)(45) = 0
Expand and simplify the equation:
12a(a - 45) + 540a + 2025 = 0
Rearrange the equation:
12a² - 540a + 2025 = 0
Now, let's solve this quadratic equation:
Using the quadratic formula: a = (-b ± √(b² - 4ac)) / (2a)
a = (-(-540) ± √((-540)² - 4(12)(2025))) / (2(12))
Simplifying:
a = (540 ± √(291600 - 97200)) / 24
a = (540 ± √194400) / 24
a = (540 ± 440) / 24
There are two possible solutions for "a":
a₁ = (540 + 440) / 24 = 980 / 24 = 40.833
a₂ = (540 - 440) / 24 = 100 / 24 = 4.167
Therefore, "a" can be approximately 40.833 or 4.167 in order for FC and FD to be perpendicular.