Question

a triangle with side lengths labeled 3, 4, and 5 (the base is 5, one lateral side is 3, and the other lateral side is 4).

Explanation:

To determine the type of triangle (e.g., right - angled) or find some property:

Step 1: Check Pythagorean theorem

For a triangle with side lengths \(a = 3\), \(b = 4\), and \(c = 5\), the Pythagorean theorem states that for a right - angled triangle, \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the longest side).
Calculate \(a^{2}+b^{2}\): \(3^{2}+4^{2}=9 + 16=25\)
Calculate \(c^{2}\): \(5^{2}=25\)

Since \(3^{2}+4^{2}=5^{2}\), the triangle is a right - angled triangle with the right angle opposite the side of length 5.

If we want to find the area of the right - angled triangle:

Step 1: Recall the formula for the area of a right - angled triangle

The area \(A\) of a right - angled triangle is given by \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, we can take the base as 3 and the height as 4 (or vice - versa).

Step 2: Calculate the area

\(A=\frac{1}{2}\times3\times4 = 6\)

If the question was to determine if it's a right - angled triangle, the answer is yes (it is a right - angled triangle). If the question was to find the area, the answer is 6.

Answer:

To determine the type of triangle (e.g., right - angled) or find some property:

Step 1: Check Pythagorean theorem

For a triangle with side lengths \(a = 3\), \(b = 4\), and \(c = 5\), the Pythagorean theorem states that for a right - angled triangle, \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the longest side).
Calculate \(a^{2}+b^{2}\): \(3^{2}+4^{2}=9 + 16=25\)
Calculate \(c^{2}\): \(5^{2}=25\)

Since \(3^{2}+4^{2}=5^{2}\), the triangle is a right - angled triangle with the right angle opposite the side of length 5.

If we want to find the area of the right - angled triangle:

Step 1: Recall the formula for the area of a right - angled triangle

The area \(A\) of a right - angled triangle is given by \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, we can take the base as 3 and the height as 4 (or vice - versa).

Step 2: Calculate the area

\(A=\frac{1}{2}\times3\times4 = 6\)

If the question was to determine if it's a right - angled triangle, the answer is yes (it is a right - angled triangle). If the question was to find the area, the answer is 6.