Slove this question step by step ---**Question 5:** **Question Stem:** Two tangents of length 7 cm meet a circle of radius 7 cm. Find the shaded area. **Diagram Description:** * **Type:** Geometric figure showing a circle and related lines. * **Elements:** * A circle with its center implied at the vertex of the angle labeled 90°. * Two radii of the circle are shown, each labeled with a length of 7 cm. * The angle between these two radii at the center is labeled as 90° and also as π/2 radians. * Two lines extend from the points where the radii meet the circle. These lines are shown meeting at a point outside the circle. * These lines are labeled as having a length of 7 cm, measured from the point where they meet outside the circle to the points where they touch the circle. These lines appear to be tangents to the circle from an external point. * A line segment connects the external endpoint of the two tangents, forming a triangle. * The triangle formed by the two tangents and the line segment connecting their external endpoints is shaded.

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In this problem, we have a circle with radius 7 centimeters. Two tangents of length 7 centimeters meet this circle. We need to find the area of the shaded triangle. Let's identify the key elements: O is the center of the circle, A and B are the points where the tangents touch the circle, and P is the point where the tangents meet. The angle between the radii OA and OB is 90 degrees. We'll solve this step by step in the following scenes. Let's examine the key properties of this figure. First, we know that a tangent to a circle is perpendicular to the radius at the point of tangency. This means that angle OAP is 90 degrees and angle OBP is 90 degrees. We're also given that the angle between the radii OA and OB is 90 degrees. These right angles are crucial for determining the shape of the quadrilateral OAPB and ultimately finding the area of the shaded triangle. Now, let's analyze the quadrilateral OAPB. We've established that angles OAP and OBP are both 90 degrees. The angle AOB is also 90 degrees. For a quadrilateral, the sum of interior angles is 360 degrees. This means the fourth angle, APB, must also be 90 degrees. Additionally, we know that OA equals OB equals 7 centimeters, and PA equals PB equals 7 centimeters. With all four angles being 90 degrees and all four sides being equal, OAPB must be a square with side length 7 centimeters. The shaded triangle PAB is exactly half of this square. Now we can calculate the area of the shaded triangle PAB. We've established that triangle PAB is a right triangle with the right angle at P. The base of this triangle is PA, which equals 7 centimeters. The height is PB, which also equals 7 centimeters. Using the formula for the area of a triangle, which is one-half times the base times the height, we get: Area equals one-half times 7 centimeters times 7 centimeters, which equals 49 divided by 2, or 24.5 square centimeters. Therefore, the area of the shaded region is 24.5 square centimeters. To summarize what we've learned: We started with a problem involving two tangents of length 7 centimeters meeting a circle of radius 7 centimeters. By analyzing the properties of tangents and radii, we discovered that the quadrilateral OAPB forms a square with side length 7 centimeters. The shaded triangle PAB is a right triangle that forms exactly half of this square. Using the formula for the area of a triangle, we calculated that the area equals one-half times the base of 7 centimeters times the height of 7 centimeters, which gives us 24.5 square centimeters. Therefore, the area of the shaded region is 24.5 square centimeters, which matches the expected answer.