this is name 75 2025-03-24 Algebra 0 views Problem Diagram Problem Statement this is des 75 Solution Process Okay, let's see what the problem is here. The user provided an image link, but I can't view images. Hmm, that's a problem. The user mentioned it's a math problem related to a right triangle with specific coordinates. Wait, the coordinates given are A(0,0), B(0,4), and C(3,0). So, plotting these points, A is at the origin, B is 4 units up the y-axis, and C is 3 units along the x-axis. That forms a right triangle at the origin. The question is asking for the area of the triangle. First, I need to recall how to find the area of a right triangle. The formula is (base * height)/2. In this case, the legs of the triangle are along the axes. The base can be the distance from A to C, which is 3 units (since x goes from 0 to 3, y remains 0). The height is the distance from A to B, which is 4 units (y from 0 to 4, x remains 0). So plugging into the formula: (3*4)/2 = 12/2 = 6. But wait, maybe the user wants a more detailed breakdown. Let me check if there's another method, like using coordinates with the shoelace formula. For coordinates (x1,y1), (x2,y2), (x3,y3), the area is |(x1(y2-y3) + x2(y3-y1) + x3(y1-y2))/2|. Plugging in the points: A(0,0), B(0,4), C(3,0). So compute 0*(4-0) + 0*(0-0) + 3*(0-4) = 0 + 0 + 3*(-4) = -12. Take absolute value and divide by 2: | -12 | /2 = 12/2=6. Same answer. So the area is 6. I should guide the student through these steps, maybe start by identifying the coordinates, then determine the base and height, apply the formula, and verify with another method if possible. Make sure they understand why the base and height are 3 and 4. Also, check if they might have confused legs or used hypotenuse by mistake. But since it's a right triangle at the origin, it's straightforward. **Step-by-Step Explanation:** 1. **Identify the Coordinates:** The vertices of the triangle are given as \( A(0,0) \), \( B(0,4) \), and \( C(3,0) \). Let’s plot these points mentally: - Point \( A \) is at the origin \((0,0)\). - Point \( B \) is 4 units up the y-axis. - Point \( C \) is 3 units along the x-axis. 2. **Determine the Base and Height:** Since the triangle is right-angled at \( A(0,0) \): - The **base** is the horizontal side \( AC \). The length of \( AC \) is \( |3 - 0| = 3 \) units. - The **height** is the vertical side \( AB \). The length of \( AB \) is \( |4 - 0| = 4 \) units. 3. **Apply the Area Formula for a Right Triangle:** The area of a right triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Plug in the values: \[ \text{Area} = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \] 4. **Verify with the Shoelace Formula (Optional):** For confirmation, use the shoelace formula for coordinates \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\): \[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \] Substitute \( A(0,0) \), \( B(0,4) \), \( C(3,0) \): \[ \text{Area} = \frac{1}{2} |0(4 - 0) + 0(0 - 0) + 3(0 - 4)| = \frac{1}{2} |0 + 0 - 12| = \frac{12}{2} = 6 \] **Final Answer:** The area of the triangle is \(\boxed{6}\). 1 Show Full Solution Final Answer Understanding the Solution Share Solution
