Question

A right triangle is shown. From A to C, a diagonal line is formed that measures 8, and from A to B, a slanted line is formed that measures 17. And from C to B, a horizontal line is formed that measures 15. Use the inverse of cosine to find the approximate interior angle measure at vertex A . Round your answer to the nearest whole degree. (1 point) A≈ °

Answer 1

To find the angle at vertex A in triangle ABC, we can use the Law of Cosines. According to the Law of Cosines:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

where \(c\) is the side opposite to angle C, and \(a\) and \(b\) are the other two sides.

In this case:

- Let \(a = AC = 8\) (the diagonal line),
- Let \(b = AB = 17\) (the slanted line),
- Let \(c = BC = 15\) (the horizontal line).

We want to find the angle at vertex A (let's call it angle A). We can rearrange the Law of Cosines formula to solve for \( \cos A \):

\[
AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos A
\]

Substituting the values:

\[
17^2 = 8^2 + 15^2 - 2 \cdot 8 \cdot 15 \cdot \cos A
\]

Calculating the squares:

\[
289 = 64 + 225 - 240 \cdot \cos A
\]

Simplifying:

\[
289 = 289 - 240 \cdot \cos A
\]

This simplifies to:

\[
0 = -240 \cdot \cos A
\]

Thus, we have:

\[
240 \cdot \cos A = 0
\]

So,

\[
\cos A = 0
\]

Now, we can find angle A using the inverse cosine:

\[
A = \cos^{-1}(0)
\]

The angle whose cosine is zero is \(90^\circ\).

Therefore, the approximate interior angle measure at vertex A is

\[
\boxed{90^\circ}.
\]