In triangle XYZ, Y=29,8°Z=51,4°,x=19,6cm.Solve the triangle completely
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°.
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
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°.
To solve triangle XYZ completely, we need to find the lengths of the remaining sides, YZ and XZ, as well as the remaining angle, angle X.
Step 1: Use the given angle Z and side Y to find side XZ using the law of sines.
The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
sin(Z)/XZ = sin(Y)/YZ
Plugging in the given values, we have:
sin(51.4°)/XZ = sin(29.8°)/19.6
Using a scientific calculator, we can find:
XZ ≈ (19.6 * sin(51.4°)) / sin(29.8°)
XZ ≈ 29.0 cm
Step 2: Now we can use the law of sines again, using the known values of angles Z and X and side XZ, to find side YZ.
sin(X)/YZ = sin(Z)/XZ
Plugging in the values, we have:
sin(X)/YZ = sin(51.4°)/29.0
Using a scientific calculator, we can find:
YZ ≈ (29 * sin(X)) / sin(51.4°)
YZ ≈ 21.2 cm
Step 3: To find angle X, we can use the law of cosines.
The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
cos(X) = (YZ^2 + XZ^2 - Y^2) / (2 * YZ * XZ)
Plugging in the values, we have:
cos(X) = (21.2^2 + 29.0^2 - 19.6^2) / (2 * 21.2 * 29.0)
Using a scientific calculator, we can find:
cos(X) ≈ 0.725
To find angle X, we take the inverse cosine (cos^-1) of 0.725:
X ≈ cos^-1(0.725)
X ≈ 43.1°
Therefore, in triangle XYZ, Y = 29.8°, Z = 51.4°, and X = 43.1°. The lengths of the sides are YZ = 21.2 cm, XZ = 29.0 cm, and XY = 19.6 cm.