A Unit Square problem

Geometry Level 3

In the figure, $ABCD$ is a unit square. A circle is drawn with centre $O$ on the extended line $CB$ and passing through $A$ . If the diagonal is tangent to the circle at $A$ , what is the area of the shaded region?

\frac {11-\pi}4

\frac {6-\pi}4

None of the others

\frac {8-\pi}4

\frac {9-\pi}4

\frac {7-\pi}4

\frac {5-\pi}4

We can't say

1 solution

Chew-Seong Cheong
Apr 14, 2019

If the circular arc $AX$ is tangent to the diagonal $AC$ , then $\angle OAC = 90^\circ$ , and $\angle OAB = \angle OAC - \angle BAC = 90^\circ - 45^\circ = 45^\circ$ . Then $AO =\sqrt 2$ and the area of the shaded area:

$\begin{aligned} [AXCD] & = [ABCD] - [ABX] \\ & = [ABCD] - \big([OAX] - [OAB]\big) \\ & = 1 \times 1 - \frac {\pi \left(\sqrt 2\right)^2}8 + \frac {1\times 1}2 \\ & = \boxed{\dfrac {6-\pi}4} \end{aligned}$

@chakravarthy b , you should have used a large figure. Put all math items $ABCD$ , $O$ , $CD$ , and $A$ in LaTex (use \ ( and \ ) --- without space between backslash \ and brackets ( )) instead of in bold. For fraction in the answer options use \frac {6-\pi}{4} or \frac {6-\pi}4, use "None of the others" instead of "None of the above", because this option may not be at the bottom. Use capital letters for "None" and "We" to start a sentence.

Chew-Seong Cheong - 2 years, 1 month ago

Chakravarthy b i am vamsi dasari. Sri venkataswara class mate.please whatsapp me 7386479513. I dont have your number. I want to talk with you and discuss with you. Hope you see this... Problem is good .please see this.

vamsi vamsi - 11 months, 3 weeks ago

A Unit Square problem

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