QuestionCalculate the distance $BC$ in triangle $ABC$ where $AB=1.5$ km, $AC=1.2$ km, and $\hat{A}=50^\circ$ .

Studdy Solution

STEP 1

Assumptions1. The triangle is $\triangle ABC$ . . The length of $AB$ is $1.5$ km.
3. The length of $AC$ is $1.$ km.
4. The angle $\hat{A}$ is $50^{\circ}$ .
5. We are asked to find the length of $BC$ .

STEP 2

We can use the Law of Cosines to find the length of $BC$ . The Law of Cosines states that for any triangle with sides of lengths $a$ , $b$ , and $c$ and an angle $\gamma$ opposite side $c$ , the following equation holds $c^2 = a^2 + b^2 -2ab\cos(\gamma)$

STEP 3

In our case, $a = AB =1.5$ km, $b = AC =1.2$ km, and $\gamma = \hat{A} =50^{\circ}$ . We want to find $c = BC$ . Substituting our values into the Law of Cosines gives us $BC^2 = (AB)^2 + (AC)^2 -2(AB)(AC)\cos(\hat{A})$

STEP 4

Substitute the given values into the equation.
$BC^2 = (1.)^2 + (1.2)^2 -2(1.)(1.2)\cos(50^{\circ})$

STEP 5

Calculate the square of $BC$ .
$BC^2 =2.25 +1.44 -3.\cos(50^{\circ})$

STEP 6

Calculate the cosine of $50^{\circ}$ .
$\cos(50^{\circ}) \approx0.6428$

STEP 7

Substitute the value of $\cos(50^{\circ})$ into the equation.
$BC^2 =2.25 +1.44 -3.6(0.642)$

STEP 8

Calculate $BC^2$ .
$BC^2 \approx0.9648$

STEP 9

To find $BC$ , we take the square root of $BC^2$ .
$BC = \sqrt{BC^2}$

STEP 10

Substitute the value of $BC^2$ into the equation.
$BC = \sqrt{0.9648}$

STEP 11

Calculate $BC$ .
$BC \approx0.982 \, km$ The distance BC is approximately0.982 km.

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