To solve the triangle completely, we need to find the lengths of side YZ and side XZ, as well as the measure of angle X.
We can start by using the Law of Sines, which states that the ratio of a side length to the sine of its opposite angle is the same for all three sides in a triangle.
Using the Law of Sines, we have:
sin(Y) / YZ = sin(Z) / XZ
Plugging in the given values, we have:
sin(29.8°) / YZ = sin(51.4°) / XZ
Now we can solve for YZ by substituting the known values:
sin(29.8°) / YZ = sin(51.4°) / XZ
sin(29.8°) / YZ = sin(51.4°) / 19.6cm
YZ = sin(29.8°) * (19.6cm / sin(51.4°))
YZ ≈ 10.5 cm
Next, we can use the Law of Sines again to solve for XZ:
sin(Y) / YZ = sin(X) / XZ
Plugging in the given values and the previously calculated value for YZ, we have:
sin(29.8°) / 10.5cm = sin(X) / XZ
Now we can solve for XZ:
XZ = sin(X) * (10.5cm / sin(29.8°))
Finally, we can use the fact that the sum of the angles in a triangle is always 180° to solve for angle X:
180° - 29.8° - 51.4° = X
X ≈ 98.8°
Therefore, the lengths of sides YZ and XZ are approximately 10.5 cm and the measure of angle X is approximately 98.8°.
In triangle XYZ, Y=29,8°Z=51,4°,x=19,6cm.Solve the triangle completely
3 answers
the usual labeling puts XY across from Z. That would make the law of sines
sin(Y) / XZ = sin(Z) / XY
or
sin(X) / YZ = sin(Y) / XZ
sin(Y) / XZ = sin(Z) / XY
or
sin(X) / YZ = sin(Y) / XZ
Apologies for the confusion. Let's re-label the triangle XYZ such that Y is opposite to angle X and Z is opposite to angle Y. The law of sines will be:
sin(X) / YZ = sin(Y) / XZ
Plugging in the given values:
sin(X) / YZ = sin(29.8°) / XZ
YZ = 19.6 cm
Y = 29.8°
Z = 51.4°
Let's solve for XZ first. Rearranging the equation:
XZ = (sin(Y) / sin(X)) * YZ
XZ = (sin(29.8°) / sin(X)) * 19.6 cm
Now we need to find sin(X). Using the fact that the sum of the angles in a triangle is 180°:
X = 180° - 29.8° - 51.4°
X ≈ 98.8°
Finally, plugging in the values:
XZ = (sin(29.8°) / sin(98.8°)) * 19.6 cm
XZ ≈ 6.05 cm
Therefore, the lengths of sides YZ and XZ are approximately 19.6 cm and 6.05 cm, and the measure of angle X is approximately 98.8°.
sin(X) / YZ = sin(Y) / XZ
Plugging in the given values:
sin(X) / YZ = sin(29.8°) / XZ
YZ = 19.6 cm
Y = 29.8°
Z = 51.4°
Let's solve for XZ first. Rearranging the equation:
XZ = (sin(Y) / sin(X)) * YZ
XZ = (sin(29.8°) / sin(X)) * 19.6 cm
Now we need to find sin(X). Using the fact that the sum of the angles in a triangle is 180°:
X = 180° - 29.8° - 51.4°
X ≈ 98.8°
Finally, plugging in the values:
XZ = (sin(29.8°) / sin(98.8°)) * 19.6 cm
XZ ≈ 6.05 cm
Therefore, the lengths of sides YZ and XZ are approximately 19.6 cm and 6.05 cm, and the measure of angle X is approximately 98.8°.