Question

In a right triangle, angle C measures 40°. The hypotenuse of the triangle is 10 inches long. What is the approximate length of the side adjacent to angle C?
6.4 inches
7.7 inches
8.4 inches
13.1 inches

Answer 1

Answer

Find out the what is the approximate length of the side adjacent to angle C .

To prove

As given

In a right triangle, angle C measures 40°.

The hypotenuse of the triangle is 10 inches long.

Than by using the trignometric identity

$cos\angle C= \frac{Base}{Hypotenuse}\\cos\angle C= \frac{BC}{AC}$

As shown the diagram is given below

AC= 10 inches , ∠C = 40 °

cos 40 = 0.766 (approx)

Put in the above formula

0.766 × 10 = BC

7.66 = BC

7.7 inches (approx) = BC

Option (b) is correct .

Answer 2

The correct option is Option C $\boxed{{\mathbf{7}}{\mathbf{.66 inches}}}$ .

Further explanation:

The cosine ratio can be represented as,

  $\cos \theta  = \frac{{{\text{base}}}}{{{\text{hypotenuse}}}}$

Here, base is the length of the side adjacent to angle $\theta$  and hypotenuse is the longest side of the right angle triangle.

The length of side opposite to angle $\theta$  is perpendicular that is used for the sine ratio.

Step by step explanation:

Step 1:

From the given information, the observed right angle is attached.

First find the hypotenuse and the base of the right angle triangle.

It can be seen from the attached figure that the side $BC$  is adjacent to angle $C$  and the side $AC$  is the hypotenuse of triangle.

Thus, the ${\text{base}}=BC$  and ${\text{hypotenuse}}=10$ .

Step 2:

We know that the cosine ratio is $\cos \theta =\frac{{{\text{base}}}}{{{\text{hypotenuse}}}}$ .

Therefore, it can be written as,

 $\cos \theta=\frac{{BC}}{{AC}}$

Now substitute the value $BC=x$  and ${\text{AC}}=10$  in the cosine ratio.

$\begin{aligned}\cos C&=\frac{x}{{10}}\\{\text{co}}s40&=\frac{x}{{10}}\\0.766&=\frac{x}{{10}}\\x&=7.66\\\end{aligned}$

Therefore, the approximate length of the side adjacent to angle $C$  is $7.7{\text{ inches}}$  .

Thus, option C is correct.

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Trigonometry

Keywords: Distance, Pythagoras theorem, base, perpendicular, hypotenuse, right angle triangle, units, squares, sum, cosine ratio, adjacent side to angle, opposite side to angle.