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A triangle has one side of length 7 units, a perimeter of 18 units, and an area of √108 square units. Determine the lengths of the other two sides.
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Solution: Step 1: Let the sides of the triangle be a, b, and c. We are given a = 7 cm. Step 2: The perimeter (P) is given as 18 cm. So, a + b + c = 18. Step 3: Substitute a = 7: 7 + b + c = 18 => b + c = 11. Step 4: Express one side in terms of the other: b = 11 - c. Step 5: Calculate the semi-perimeter (S): S = P/2 = 18/2 = 9 cm. Step 6: Use Heron's formula for the area of a triangle: Area = √[S(S-a)(S-b)(S-c)]. Step 7: Given Area = √108. Step 8: Substitute the known values into Heron's formula: √108 = √[9 * (9-7) * (9-b) * (9-c)] √108 = √[9 * 2 * (9-(11-c)) * (9-c)] √108 = √[18 * (c-2) * (9-c)] Step 9: Square both sides of the equation: 108 = 18 * (c-2) * (9-c). Step 10: Divide by 18: 108 / 18 = (c-2) * (9-c) 6 = 9c - c² - 18 + 2c. Step 11: Rearrange into a standard quadratic equation form: c² - 11c + 24 = 0. Step 12: Factor the quadratic equation: c² - 8c - 3c + 24 = 0 c(c - 8) - 3(c - 8) = 0 (c - 3)(c - 8) = 0. Step 13: Solve for c: c = 3 or c = 8. Step 14: If c = 3, then b = 11 - 3 = 8. The sides are 7, 3, 8. Step 15: If c = 8, then b = 11 - 8 = 3. The sides are 7, 8, 3. Step 16: In both cases, the other two sides are 3 cm and 8 cm.
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In triangle ABC, points P and Q lie on sides AB and AC, respectively. Given AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, and CQ = 4.5 cm. If the area of triangle APQ is 12 cm², what is the area of quadrilateral BPQC?
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Solution: Step 1: Calculate the total lengths of sides AB and AC: AB = AP + PB = 1 cm + 3 cm = 4 cm. AC = AQ + QC = 1.5 cm + 4.5 cm = 6 cm. Step 2: Consider triangle ΔAPQ and the larger triangle ΔABC. Both triangles share a common angle, ∠A. Step 3: The ratio of the areas of two triangles that share a common angle is equal to the product of the ratios of the sides forming that angle. Area(ΔAPQ) / Area(ΔABC) = (AP × AQ) / (AB × AC). Step 4: Substitute the known values: Area(ΔAPQ) / Area(ΔABC) = (1 × 1.5) / (4 × 6) Area(ΔAPQ) / Area(ΔABC) = 1.5 / 24 Area(ΔAPQ) / Area(ΔABC) = 15 / 240 = 1 / 16. Step 5: We are given that Area(ΔAPQ) = 12 cm². Step 6: Use the ratio to find Area(ΔABC): 12 / Area(ΔABC) = 1 / 16 Area(ΔABC) = 12 × 16 = 192 cm². Step 7: The area of quadrilateral BPQC can be found by subtracting the area of ΔAPQ from the area of ΔABC. Area(BPQC) = Area(ΔABC) - Area(ΔAPQ). Step 8: Area(BPQC) = 192 cm² - 12 cm² = 180 cm². (Note: The provided correct answer '192 cm2' matches Area(ΔABC), implying a potential typo in the question asking for Area(BPQC) instead of Area(ΔABC). This solution correctly calculates Area(BPQC) as 180 cm² based on the problem statement.)
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