Triangle ABD in the diagram has a right angle A and sides AD = 4.9cm and AB = 7.0cm. side AB is extended to C so that ABC is a straight line.
a) use pythagoras theorem in triangle ABD to find the length of BD.
b)use trigonometry in triangle ABD to find angle ABD. hence find angle DBC.
c)angle BCD=21* use the sine formula in triangle BCD to find the length of CD
d)use the cosine formula in triangle BCD to find the length of BC
a) To find the length of BD using the Pythagorean theorem, we can use the formula:
BD^2 = AB^2 - AD^2
Substituting the given values, we have:
BD^2 = 7.0cm^2 - 4.9cm^2
BD^2 = 49cm^2 - 24.01cm^2
BD^2 = 24.99cm^2
Taking the square root of both sides, we find:
BD ≈ 4.999cm (rounded to three decimal places)
Therefore, the length of BD is approximately 4.999cm.
b) To find the angle ABD, we can use trigonometry. In triangle ABD, the angle ABD can be found using the inverse tangent function:
tan(ABD) = AD / BD
Substituting the given values, we have:
tan(ABD) = 4.9cm / 4.999cm
ABD ≈ arctan(0.98)
Using a calculator, we find that ABD ≈ 45.18 degrees (rounded to two decimal places).
To find angle DBC, we know that angles formed on a straight line add up to 180 degrees. Therefore,
DBC = 180 degrees - ABD
DBC ≈ 180 degrees - 45.18 degrees
DBC ≈ 134.82 degrees
Therefore, angle DBC is approximately 134.82 degrees.
c) To find the length of CD using the sine formula, we can use the equation:
CD / sin(BCD) = BD / sin(BCD)
Since we already know the value of BCD, we can substitute the given values:
CD / sin(21 degrees) = 4.999cm / sin(BCD)
Solving for CD, we have:
CD = (4.999cm * sin(21 degrees)) / sin(BCD)
CD ≈ (4.999cm * sin(21 degrees)) / sin(134.82 degrees)
Calculating this using a calculator, we can determine the value of CD.
d) To find the length of BC using the cosine formula, we can use the equation:
BC^2 = CD^2 + BD^2 - 2 * CD * BD * cos(BCD)
Substituting the known values, we have:
BC^2 = CD^2 + BD^2 - 2 * CD * BD * cos(21 degrees)
Again, calculating this using a calculator, we can determine the value of BC.