Answer:
The measure of ∠BAC is 30 degrees.
Explanation:
The given figure shows a right triangle ABC, where ∠ACB is a right angle. Point D is on AB such that AD = 6 cm and DB = 4 cm. Point E is on BC such that BE = 5 cm.
To find:
a) The length of AC.
b) The length of DE.
c) The measure of ∠BAC in degrees.
Solution:
a) Using the Pythagorean theorem in triangle ABC,
AC² = AB² + BC²
AC² = (AD + DB)² + BE²
AC² = (6 + 4)² + 5²
AC² = 100
AC = √100
AC = 10 cm
Therefore, the length of AC is 10 cm.
b) To find the length of DE, we can use the similarity of triangles ADE and ABC.
Since ∠AED = ∠ACB, we have:
ADE ~ ABC
Therefore, we can write:
DE/BC = AD/AB
Substituting the given values, we get:
DE/5 = 6/(6 + 4)
DE/5 = 6/10
DE = (6/10) × 5
DE = 3 cm
Therefore, the length of DE is 3 cm.
c) To find the measure of ∠BAC in degrees, we can use the sine function in triangle ABC.
sin(∠BAC) = opposite/hypotenuse
sin(∠BAC) = BC/AC
sin(∠BAC) = 5/10
sin(∠BAC) = 1/2
Taking the inverse sine of both sides, we get:
∠BAC = sin⁻¹(1/2)
∠BAC = 30°
Therefore, the measure of ∠BAC is 30 degrees.
(Reference to chatgpts work)