The length of AD is approximately 8.4 cm.
Identify relevant relationships:
We are given:
BC = 6 cm (length of hypotenuse in triangle ABC)
tan x = 1.3 (tangent of angle x in triangle ABC)
cos y = 0.4 (cosine of angle y in triangle BCD)
We need to find AD (length of a side in triangle ABD).
Use trigonometric relationships:
Tan x = 1.3: This tells us the ratio of the opposite side (AC) to the adjacent side (BC) in triangle ABC. We can write it as AC/BC = 1.3.
cos y = 0.4: This tells us the ratio of the adjacent side (CD) to the hypotenuse (BC) in triangle BCD. We can write it as CD/BC = 0.4.
Relate triangles ABC and BCD:
Since angle ACB is a right angle and BCD is a straight line, we can see that triangles ABC and BCD are right-angled triangles. Additionally, angle x and angle y are complementary angles (they add up to 90 degrees).
Solve for AC:
From the tan x equation, we can write AC = 1.3 * BC. Substituting BC = 6 cm, we get AC = 1.3 * 6 cm = 7.8 cm.
Solve for CD:
From the cos y equation, we can write CD = 0.4 * BC. Substituting BC = 6 cm, we get CD = 0.4 * 6 cm = 2.4 cm.
Find AD:
Finally, notice that AD is the hypotenuse of triangle ABD, and its two legs are AC and CD. We can use the Pythagorean theorem:
AD² = AC² + CD²
Substituting the values we found:
AD² = (7.8 cm)² + (2.4 cm)²
AD² ≈ 64.84 cm² + 5.76 cm² ≈ 70.6 cm²
Taking the square root of both sides:
AD ≈ √70.6 cm² ≈ 8.4 cm
Therefore, the length of AD is approximately 8.4 cm.