+363 In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 2$, $BY = 16$, $\angle ACB = 90^\circ$, and $CY = 2 \cdot CX$, then find $AB$.
+400 Certainly, let's find the length of AB in rectangle WXYZ.
1. Define Variables
Let's denote:
WX = YZ = length = 'l'
XY = WZ = width = 'w'
CX = 'x'
CY = 2x
2. Utilize Given Information
AX = 2
BY = 16
∠ACB = 90°
3. Analyze Triangle ACB
Since ∠ACB = 90°, triangle ACB is a right-angled triangle.
We can apply the Pythagorean Theorem:
AC^2 + BC^2 = AB^2
4. Express AC and BC in terms of other variables
AC = AX + CX = 2 + x
BC = BY - CY = 16 - 2x
5. Substitute into the Pythagorean Theorem
(2 + x)^2 + (16 - 2x)^2 = AB^2
4 + 4x + x^2 + 256 - 64x + 4x^2 = AB^2
5x^2 - 60x + 260 = AB^2
6. Find the Relationship between Length and Width
In right-angled triangle ACB:
tan(∠CAB) = BC/AC
tan(∠CAB) = (16 - 2x) / (2 + x)
In right-angled triangle CXY:
tan(∠CXY) = CY/CX
tan(∠CXY) = 2x / x = 2
Since XY || WZ, ∠CAB = ∠CXY
Therefore, (16 - 2x) / (2 + x) = 2
7. Solve for 'x'
16 - 2x = 2(2 + x)
16 - 2x = 4 + 2x
12 = 4x
x = 3
8. Calculate AB
Substitute the value of 'x' in the equation for AB^2:
AB^2 = 5(3)^2 - 60(3) + 260
AB^2 = 45 - 180 + 260
AB^2 = 125
AB = √125
AB = 5√5
Therefore, the length of AB is 5√5.
