Question 2---Here is the extracted content from the image, presented in a structured plain text format: --- **Question 1** **Textual Information:** * **Question Stem (Incomplete):** "ght of the triangle below and then find a formula for its area in te" * *Note:* The full question likely starts with "Find the hei..." and ends with "terms of..." **Chart/Diagram Description:** * **Type:** Geometric figure, specifically a right-angled triangle. * **Main Elements:** * **Shape:** A triangle with three sides and three vertices. * **Angles:** One angle at the top-left vertex is marked with a square symbol, indicating it is a right angle (90 degrees). Another acute angle, located at the bottom-left vertex, is labeled 'θ' (theta). * **Sides:** The side opposite the right angle (the hypotenuse) is labeled with the variable 'x'. * **Orientation:** The triangle is oriented such that the base appears horizontal, and the right angle is at the top corner where the vertical side (height) meets the top segment of the base. The side 'x' is the slanted side. --- **Question 2** **Textual Information:** * **Question Stem (Incomplete):** * "...meters long runs from the top of a pole to the ground as shown" * "...s an angle of 3k° with the ground. Calculate the height of the pole." * *Note:* This appears to be a separate word problem. The first sentence is missing its beginning, likely describing what is "meters long" (e.g., a rope, wire, or ladder). The second sentence is missing its subject. "3k°" is extracted as seen, assuming 'k' is a variable or part of an unknown angle value. **Chart/Diagram Description:** * No chart or diagram is provided for this question; it is a text-based problem description. ---

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Let's analyze this right triangle problem. We have a right triangle with hypotenuse x and angle theta at the bottom left vertex. Our goal is to find the height and derive the area formula. The triangle has a right angle at the top, and we can identify the height as the vertical side and the base as the horizontal side. To find the height of the triangle, we use the sine trigonometric ratio. Sine theta equals the opposite side divided by the hypotenuse. In our triangle, the height h is the side opposite to angle theta, and x is the hypotenuse. Therefore, sine theta equals h over x. Solving for h, we get h equals x sine theta. This gives us the height in terms of the hypotenuse and the angle. Now let's find the base of the triangle using the cosine ratio. Cosine theta equals the adjacent side divided by the hypotenuse. In our triangle, the base b is the side adjacent to angle theta, and x is the hypotenuse. Therefore, cosine theta equals b over x. Solving for b, we get b equals x cosine theta. This gives us the base length in terms of the hypotenuse and the angle. Now let's derive the area formula for our triangle. The area of any triangle is one half times base times height. Substituting our expressions, we get area equals one half times x cosine theta times x sine theta. This simplifies to one half x squared sine theta cosine theta. Using the double angle identity, sine of two theta equals two sine theta cosine theta, we can write the final area formula as one quarter x squared sine of two theta. Let's solve the triangle problem step by step. We have a right triangle with hypotenuse x and angle theta. To find the height, we use sine: height equals x sine theta. For the base, we use cosine: base equals x cosine theta. The area formula is one-half base times height, which simplifies to one-quarter x squared sine of two theta using the double angle identity. Now let's solve the pole problem. We have a wire of length L meters running from the top of a pole to the ground, making an angle of three k degrees with the ground. To find the pole height, we use the sine function. Since sine equals opposite over hypotenuse, we have sine of three k degrees equals h over L. Therefore, the pole height h equals L times sine of three k degrees.