AT is a tangent to the given circle. OA = 13 cm and the circle has radius of 5 cm. Find the perimeter of the shaded region.

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QUESTION 5

Given:
AT is a tangent to the circle at T.
OA = 13 cm.
Radius of the circle (OT) = 5 cm.

Since AT is a tangent to the circle at T, the radius OT is perpendicular to the tangent AT. Therefore, $\triangle OAT$ is a right-angled triangle with the right angle at T.

Assuming the "shaded region" refers to the region bounded by the tangent segment AT, the line segment AP (where P is the point where OA intersects the circle), and the arc TP.

Step 1: Calculate the length of the tangent segment AT.
Using the Pythagorean theorem in $\triangle OAT$:

$$OA^2 = OT^2 + AT^2$$ $$13^2 = 5^2 + AT^2$$ $$169 = 25 + AT^2$$ $$AT^2 = 169 - 25$$ $$AT^2 = 144$$ $$AT = \sqrt{144} = 12 \text{ cm}$$

Step 2: Calculate the length of the segment AP.
Point P is on the circle and on the line segment OA. OP is a radius of the circle.

$$OP = \text{radius} = 5 \text{ cm}$$ $$AP = OA - OP$$ $$$$