In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, what is the area of t

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Question: In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, what is the area of the triangle?

Options:

Correct Answer: 96 cm²

Solution:

Using Heron\'s formula, the semi-perimeter s = (7 + 24 + 25)/2 = 28. Area = √[s(s-a)(s-b)(s-c)] = √[28(28-7)(28-24)(28-25)] = √[28*21*4*3] = 84 cm².

In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, what is the area of the triangle?

In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, what is the area of the triangle?

Correct Answer: 84 cm²

Step 1: Identify the lengths of the sides of triangle ABC. We have AB = 7 cm, AC = 24 cm, and BC = 25 cm.
Step 2: Calculate the semi-perimeter (s) of the triangle using the formula s = (AB + AC + BC) / 2.
Step 3: Substitute the values into the formula: s = (7 + 24 + 25) / 2 = 28 cm.
Step 4: Use Heron's formula to find the area of the triangle. The formula is Area = √[s(s-a)(s-b)(s-c)], where a, b, and c are the lengths of the sides.
Step 5: Substitute the values into Heron's formula: Area = √[28(28-7)(28-24)(28-25)].
Step 6: Calculate each part: (28-7) = 21, (28-24) = 4, (28-25) = 3.
Step 7: Now substitute these values back into the formula: Area = √[28 * 21 * 4 * 3].
Step 8: Calculate the product: 28 * 21 = 588, then 588 * 4 = 2352, and finally 2352 * 3 = 7056.
Step 9: Take the square root of 7056 to find the area: Area = √7056 = 84 cm².

Heron's Formula – A method to calculate the area of a triangle when the lengths of all three sides are known.
Semi-perimeter – The semi-perimeter of a triangle is half the sum of its side lengths, used in Heron's formula.
Triangle Inequality Theorem – A principle that states the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.