In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?

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Question: In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?

Options:

Correct Answer: 84

Solution:

Using Heron\'s formula, 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.

In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?

In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?

Step 1: Identify the sides of the triangle. Here, a = 7, b = 24, and c = 25.
Step 2: Calculate the semi-perimeter (s) of the triangle using the formula s = (a + b + c) / 2.
Step 3: Substitute the values into the formula: s = (7 + 24 + 25) / 2 = 28.
Step 4: Use Heron's formula to find the area: Area = √[s(s-a)(s-b)(s-c)].
Step 5: Calculate (s-a), (s-b), and (s-c): (28-7) = 21, (28-24) = 4, (28-25) = 3.
Step 6: Substitute these values into the area formula: Area = √[28 * 21 * 4 * 3].
Step 7: Calculate the product inside the square root: 28 * 21 = 588, then 588 * 4 = 2352, and finally 2352 * 3 = 7056.
Step 8: Find the square root of 7056 to get the area: Area = √7056 = 84.

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 (s) is half the sum of the triangle's 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.