Question

what is the measure of the smallest angle of the triangle?
15
18
?°
26
round to the nearest hundredth.

Explanation:

To find the smallest angle of a triangle when we know the lengths of all three sides, we can use the Law of Cosines. The smallest angle is opposite the shortest side. In this triangle, the sides are \( a = 15 \), \( b = 18 \), and \( c = 26 \). The shortest side is \( a = 15 \), so the angle opposite to it (let's call it \( \alpha \)) is the smallest angle.

The Law of Cosines formula is:

$$ \cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc} $$

Step 1: Substitute the values into the formula

We substitute \( a = 15 \), \( b = 18 \), and \( c = 26 \) into the formula:

$$ \cos(\alpha) = \frac{18^2 + 26^2 - 15^2}{2 \times 18 \times 26} $$

Step 2: Calculate the numerator and the denominator

First, calculate the squares:
\( 18^2 = 324 \)
\( 26^2 = 676 \)
\( 15^2 = 225 \)

Now, calculate the numerator:
\( 324 + 676 - 225 = 1000 - 225 = 775 \)

Next, calculate the denominator:
\( 2 \times 18 \times 26 = 36 \times 26 = 936 \)

So,

$$ \cos(\alpha) = \frac{775}{936} \approx 0.8280 $$

Step 3: Find the angle

To find \( \alpha \), we take the inverse cosine (arccos) of \( 0.8280 \):

$$ \alpha = \arccos(0.8280) $$

Using a calculator, we find:

$$ \alpha \approx 34.12^\circ $$

Answer:

The measure of the smallest angle is approximately \( \boxed{34.12^\circ} \) (rounded to the nearest hundredth).