Question

A triangle has sides of 10, 9, 13 cm. Can it be a right triangle? Prove

Answer 1

To determine whether the triangle with sides of 10 cm, 9 cm, and 13 cm is a right triangle, we can use the Pythagorean theorem. This theorem states that for a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the triangle is a right triangle if:

\[
c^2 = a^2 + b^2.
\]

First, we identify the longest side of the triangle. In this case, the sides are 10 cm, 9 cm, and 13 cm. The longest side is 13 cm.

Next, we will assign the sides as follows:

- \( a = 9 \) cm
- \( b = 10 \) cm
- \( c = 13 \) cm

Now we will check the Pythagorean theorem:

1. Calculate \( c^2 \):
\[
c^2 = 13^2 = 169.
\]

2. Calculate \( a^2 + b^2 \):
\[
a^2 + b^2 = 9^2 + 10^2 = 81 + 100 = 181.
\]

3. Now, we compare \( c^2 \) with \( a^2 + b^2 \):
\[
169 \neq 181.
\]

Since \( c^2 \) is not equal to \( a^2 + b^2 \), the triangle with sides 10 cm, 9 cm, and 13 cm does not satisfy the condition for being a right triangle.

Thus, we conclude that the triangle cannot be a right triangle.