This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for questions 6, 7, and 8 based on the provided image. First, let's analyze the given information and the diagram: The figure ABCD is a rectangle. EF = 2 cm. DE = 3 cm. E is the intersection point of the diagonals AC and BD. EF AB. Properties of a rectangle: • Diagonals are equal in length and bisect each other. Therefore, AE = BE = CE = DE. Since DE = 3 cm, we have AE = BE = CE = 3 cm. • All angles are 90^. So, DAB = 90^. Consider ABD: E is the midpoint of the hypotenuse BD (since BE = DE = 3 cm). EF is drawn from E such that EF AB. Since AD AB (because ABCD is a rectangle), EF must be parallel to AD. By the midpoint theorem (or intercept theorem), if a line segment passes through the midpoint of one side of a triangle and is parallel to another side, it bisects the third side. Thus, F is the midpoint of AB. Also, EF = (1)/(2) AD. Given EF = 2 cm, we can find AD: 2 = (1)/(2) AD AD = 2 × 2 = 4 cm. Since ABCD is a rectangle, BC = AD = 4 cm. Now we have: AD = 4 cm BE = 3 cm EF = 2 cm F is the midpoint of AB. 6. BF এর দৈর্ঘ্য কত সে.মি.? (What is the length of BF in cm?) Step 1: Identify the relevant triangle. Consider the right-angled triangle EFB, where EFB = 90^. Step 2: Apply the Pythagorean theorem. We know BE = 3 cm (hypotenuse) and EF = 2 cm (one leg). We need to find BF (the other leg). BE^2 = EF^2 + BF^2 3^2 = 2^2 + BF^2 9 = 4 + BF^2 BF^2 = 9 - 4 BF^2 = 5 BF = sqrt(5) সে.মি. The correct option is খ) sqrt(5). 7. AB কত সে.মি.? (What is the length of AB in cm?) Step 1: Use the property that F is the midpoint of AB. We found that F is the midpoint of AB. Therefore, AB = 2 × BF. Step 2: Calculate AB. Using the value of BF from question 6: AB = 2 × sqrt(5) AB = 2sqrt(5) সে.মি. The correct option is খ) 2sqrt(5). 8. ABCD এর ক্ষেত্রফল কত বর্গসে.মি.? (What is the area of ABCD in sq cm?) Step 1: Recall the formula for the area of a rectangle. The area of a rectangle is given by length × width. In this case, AB × AD. Step 2: Substitute the values and calculate the area. We found AB = 2sqrt(5) cm and AD = 4 cm. Area of ABCD = AB × AD Area of ABCD = (2sqrt(5)) × 4 Area of ABCD = 8sqrt(5) বর্গসে.মি. The correct option is ক) 8sqrt(5). The final answers are: 6. খ) sqrt(5) 7. খ) 2sqrt(5) 8. ক) 8sqrt(5) That's 2 down. 3 left today — send the next one.