Question

1. label each triangle with hypotenuse and adjacent.
2. identify the given ratios for each triangle.

for the left triangle (sides 3, 4, 5, right - angled):
$\frac{opposite}{hypotenuse} = $
$\frac{adjacent}{hypotenuse} = $
$\frac{opposite}{adjacent} = $
for the right triangle (with side lengths 39, 89, and a right angle):
$\frac{opposite}{hypotenuse} = $
$\frac{adjacent}{hypotenuse} = $
$\frac{opposite}{adjacent} = $

Explanation:

Step1: Label sides for left triangle

For the marked acute angle:

• Opposite side = 3, Adjacent side = 4, Hypotenuse = 5

Step2: Compute sine ratio (left)

$\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$

Step3: Compute cosine ratio (left)

$\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}$

Step4: Compute tangent ratio (left)

$\frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$

Step5: Find missing side (right triangle)

Use Pythagorean theorem: $c^2=a^2+b^2$
$$\text{Missing side} = \sqrt{89^2 - 39^2} = \sqrt{(89-39)(89+39)} = \sqrt{50 \times 128} = \sqrt{6400} = 80$$

Step6: Label sides for right triangle

For the marked acute angle:

• Opposite side = 39, Adjacent side = 80, Hypotenuse = 89

Step7: Compute sine ratio (right)

$\frac{\text{opposite}}{\text{hypotenuse}} = \frac{39}{89}$

Step8: Compute cosine ratio (right)

$\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{80}{89}$

Step9: Compute tangent ratio (right)

$\frac{\text{opposite}}{\text{adjacent}} = \frac{39}{80}$

Answer:

Left Triangle:

$\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$
$\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}$
$\frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$

Right Triangle:

$\frac{\text{opposite}}{\text{hypotenuse}} = \frac{39}{89}$
$\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{80}{89}$
$\frac{\text{opposite}}{\text{adjacent}} = \frac{39}{80}$