Math / Geometry

Question $\text{Given a triangle with side lengths } a = 500 \text{ mm}, b = 400 \text{ mm}, \text{ and } c = 375 \text{ mm, find the angles of the triangle.}$

Studdy Solution

STEP 1

1. We are given a triangle with side lengths $a = 500$ mm, $b = 400$ mm, and $c = 375$ mm.
2. We need to find the angles of the triangle.
3. We will use the Law of Cosines to find the angles.

STEP 2

1. Use the Law of Cosines to find one angle.
2. Use the Law of Cosines to find a second angle.
3. Use the sum of angles in a triangle to find the third angle.

STEP 3

To find angle $A$ , use the Law of Cosines:
$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
Substitute the given values:
$\cos A = \frac{400^2 + 375^2 - 500^2}{2 \cdot 400 \cdot 375}$
Calculate:
$\cos A = \frac{160000 + 140625 - 250000}{300000}$ $\cos A = \frac{50625}{300000}$ $\cos A = 0.16875$
Find $A$ using the inverse cosine function:
$A = \cos^{-1}(0.16875)$

STEP 4

To find angle $B$ , use the Law of Cosines:
$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$
Substitute the given values:
$\cos B = \frac{500^2 + 375^2 - 400^2}{2 \cdot 500 \cdot 375}$
Calculate:
$\cos B = \frac{250000 + 140625 - 160000}{375000}$ $\cos B = \frac{230625}{375000}$ $\cos B = 0.615$
Find $B$ using the inverse cosine function:
$B = \cos^{-1}(0.615)$

STEP 5

To find angle $C$ , use the sum of angles in a triangle:
$A + B + C = 180^\circ$
Solve for $C$ :
$C = 180^\circ - A - B$
Substitute the values of $A$ and $B$ calculated in the previous steps to find $C$ .
The angles of the triangle are approximately:
$A \approx \cos^{-1}(0.16875)$ $B \approx \cos^{-1}(0.615)$ $C = 180^\circ - A - B$

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