Let ABCD be a rectangle such that AB=5 and BC=12. There exist two distinct points X1 and X2 on BC such that ∠AX1D=∠AX2D=90∘. Suppose that d is the distance from X1 to X2. What is d2?
To find the value of d², we need to determine the distance between points X1 and X2.
Let's first draw a diagram to visualize the situation:
A ------- B
| |
| |
| X1X2 |
| |
| |
D ------- C
In the given rectangle ABCD, AB = 5 and BC = 12. We are told that there are two distinct points, X1 and X2, on BC such that ∠AX1D = ∠AX2D = 90°. Our task is to find the value of d², which represents the square of the distance between X1 and X2.
Since ∠AX1D = ∠AX2D = 90°, we can conclude that X1X2 is the diameter of the circle with AD as its chord. This circle can be constructed by drawing a perpendicular bisector to AD passing through X1 and X2.
Now, let's consider the triangle AX1D:
A ------------------ B
/ \
X1 \
\ \
D - - - - - - - - - - C
In this right triangle, AX1 = AD and X1D = AB. Therefore, AX1D is an isosceles right triangle. By the Pythagorean theorem, we can find the length of AX1 (which is equal to AD) using:
AD² = AX1² + X1D²
Since we are given that AB = 5, we can substitute the known values into the equation:
AD² = AX1² + X1D²
AD² = 5² + AB²
AD² = 25 + 5²
AD² = 25 + 25
AD² = 50
Now, we know that AD² is 50. Since AD is the diameter of the circle, the distance between X1 and X2 (d) is half the length of AD. Therefore:
d = AD / 2
d = √50 / 2
d = √25 * 2 / 2
d = √(25 * 2) / 2
d = √50 / 2
d = 5√2 / 2
To find d², we can square the value of d:
d² = (5√2 / 2)²
d² = (5 / √2)²
d² = (5² / (√2)²)
d² = (25 / 2)
d² = 25/2
Hence, d² is equal to 25/2 or 12.5.