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. Pythagoras theorem
BD²=AD²+AB²
AD and AB are known.
Solve for BD.
b. Tan(∠ABD) = opposite/adjacent = DA/AB
Solve for ∠ABD.
Since ∠ABD and ∠DBC form a straight line, they are supplementary (add up to 180°).
Solve for ∠DBC:
∠DBC = 180 - ∠ABD
c. Sine rule
The sine rule applies when all angles are known plus one side, or two sides are known and one angle which is not the included angle.
The given triangle applies in to the first case.
BD, ∠DBC, ∠BCD are known, so apply the sine rule:
a/sin(A)=b/sin(B)=c/sin(C)
where A is the angle opposite side a, etc.
BD/sin(∠BCD)=DC/sin(∠DBC)
Solve for DC.
d. The cosine rule applies when two adjacent sides and the included angle are known.
By now, BD, DC and ∠BDC are known.
Note: ∠BDC = 180-(∠DBC+∠BCD)
So apply the cosine rule to solve for c
c²=a²+a²-2abcos(C)
or
BC²=BD²+CD²-2 BD CD cos(∠BDC)
Post to check your results if you wish.
thank u
Triangle ABC is similar to triangle APQ with angle CBA equal to angle QPA and angle A common to both triangles. the length of sides AB, AP and PQ are 90cm,60cm and 30 cm. calculate the length of the side BC.
a) To find the length of BD using the Pythagorean theorem in triangle ABD, we can use the formula:
BD^2 = AB^2 - AD^2
Substituting the given values, we get:
BD^2 = (7.0cm)^2 - (4.9cm)^2
Simplifying:
BD^2 = 49.0cm^2 - 24.01cm^2
BD^2 = 24.99cm^2
Taking the square root of both sides:
BD = √(24.99cm^2)
Therefore, the length of BD is approximately 4.999 cm.
b) To find angle ABD using trigonometry in triangle ABD, we can use the tangent function:
tan(Angle ABD) = AD/AB
Substituting the given values, we get:
tan(Angle ABD) = 4.9cm / 7.0cm
Next, we can take the arctangent of both sides to find the angle ABD:
Angle ABD = arctan(4.9cm / 7.0cm)
Using a calculator, we find that Angle ABD is approximately 34.5 degrees.
Since ABC is a straight line, we know that angles ABD and DBC are supplementary, meaning they add up to 180 degrees. Therefore, angle DBC is:
Angle DBC = 180 degrees - Angle ABD
Substituting the calculated value, we get:
Angle DBC = 180 degrees - 34.5 degrees
Angle DBC is approximately 145.5 degrees.
c) To find the length of CD using the sine formula in triangle BCD, we can use the formula:
sin(angle BCD) = CD / BD
Substituting the given value of angle BCD as 21 degrees, we get:
sin(21 degrees) = CD / BD
Next, we can rearrange the formula to solve for CD:
CD = sin(21 degrees) * BD
Substituting the calculated value of BD from part a), we can find CD:
CD = sin(21 degrees) * 4.999 cm
Using a calculator, we find that CD is approximately 1.827 cm.
d) To find the length of BC using the cosine formula in triangle BCD, we can use the formula:
BC^2 = CD^2 + BD^2 - 2 * CD * BD * cos(angle BCD)
Substituting the given values, including the calculated values of CD and BD, we get:
BC^2 = (1.827 cm)^2 + (4.999 cm)^2 - 2 * 1.827 cm * 4.999 cm * cos(21 degrees)
Simplifying and calculating using a calculator, we find:
BC^2 ≈ 188.045 cm^2
Taking the square root of both sides, we get:
BC ≈ √(188.045 cm^2)
Therefore, the length of BC is approximately 13.696 cm.