Question

Given: ABC is a right triangle with right angle C. AC=15 centimeters and m∠A=40∘ . What is BC ? Enter your answer, rounded to the nearest tenth, in the box

Answer 1

12.6

Explanation:

The other answer next to mine has a great explanation, except it says round to the nearest tenth, not the nearest whole number. The correct answer would be 12.6.

Answer 2

In order to answer this question, the figure in the first picture will be helpful to understand what a right triangle is. Here, a right angle refers to
`90\°`.

However, if we want to solve the problem we have to know certain things before:

In the second figure is shown a general right triangle with its three sides and another given angle, we will name it
`\alpha`:

• The side opposite to the right angle is called The Hypotenuse (h)

• The side opposite to the angle
  `\alpha` is called the Opposite (O)

• The side next to the angle
  `\alpha` is called the Adjacent (A)

So, going back to the triangle of our question (first figure):

• The Hypotenuse is AB

• The Opposite is BC

• The Adjacent is AC

Now, if we want to find the length of each side of a right triangle, we have to use the Pythagorean Theorem and Trigonometric Functions:

Pythagorean Theorem

`h^(2)=A^(2) +O^(2)` (1)

Trigonometric Functions (here are shown three of them):

Sine:
`sin(\alpha)=(O)/(h)` (2)

Cosine:
`cos(\alpha)=(A)/(h)` (3)

Tangent:
`tan(\alpha)=(O)/(A)` (4)

In this case the function that works for this problem is cosine (3), let’s apply it here:

`cos(40\°)=(AC)/(h)`

`cos(40\°)=(15)/(h)` (5)

And we will use the Pythagorean Theorem to find the hypotenuse, as well:

`h^(2)=AC^(2)+BC^(2)`

`h^(2)=15^(2)+BC^(2)` (6)

`h=√(225+BC^2)` (7)

Substitute (7) in (5):

`cos(40\°)=(15)/(√(225+BC^2))`

Then clear BC, which is the side we want:

`{√(225+BC^2)}=(15)/(cos(40\°))`

`{{√(225+BC^2)}^2={((15)/(cos(40\°)))}^2`

`225+BC^(2)=\frac{225}{{(cos(40\°))}^2}`

`BC^2=\frac{225}{{(cos(40\°))}^2}-225`

`BC=√(158,41)`

`BC=12.58`

Finally
`BC` is approximately 13 cm