Triangle PEA has a 20-degree angle at P and a 120-degree angle at E, and the length of side EA is 6 inches. Find the lengths of the other two sides of this triangle.
Well, since the angle at P is 20 degrees, and the angle at E is 120 degrees, we can conclude that the remaining angle at A is 40 degrees. Now, let's focus on finding the lengths of sides PA and PE.
Using the Law of Sines, we can say that:
PA/sin(40) = EA/sin(20)
PA/sin(40) = 6/sin(20)
Solving for PA, we get:
PA = (6 * sin(40)) / sin(20)
PA ≈ 8.168 inches
Now, we can determine the length of side PE by using the Law of Sines again:
PE/sin(120) = EA/sin(20)
PE/sin(120) = 6/sin(20)
Solving for PE, we get:
PE = (6 * sin(120)) / sin(20)
PE ≈ 13.416 inches
So, the lengths of the other two sides of triangle PEA are approximately 8.168 inches and 13.416 inches.
To find the lengths of the other two sides of triangle PEA, we can use 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 for all sides and angles of the triangle.
Let's label the length of side PE as x and the length of side PA as y.
Using the law of sines, we can set up the following equation:
sin(20 degrees) / x = sin(120 degrees) / 6
Now we can solve for x:
x = sin(20 degrees) / (sin(120 degrees) / 6)
x = 6 * sin(20 degrees) / sin(120 degrees)
Using a calculator, we can evaluate sin(20 degrees) and sin(120 degrees):
sin(20 degrees) ≈ 0.342
sin(120 degrees) ≈ 0.866
Substituting the values:
x = 6 * 0.342 / 0.866
x ≈ 2.37 inches (rounded to two decimal places)
Now we can find the length of side PA (y) using the law of sines:
sin(120 degrees) / y = sin(20 degrees) / 6
y = 6 * sin(120 degrees) / sin(20 degrees)
Using the same values for sin(120 degrees) and sin(20 degrees) as before:
y = 6 * 0.866 / 0.342
y ≈ 15.76 inches (rounded to two decimal places)
Therefore, the length of side PE is approximately 2.37 inches, and the length of side PA is approximately 15.76 inches.
To find the lengths of the other two sides of triangle PEA, we can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all sides and angles.
Let's label the other two sides of the triangle as side PE and side PA. We know that angle E is 120 degrees, and angle P is 20 degrees. We also know the length of side EA is 6 inches.
Using the Law of Sines, we can write the following equation:
sin(angle P)/side PA = sin(angle E)/side EA
Plugging in the values we know, we get:
sin(20°)/side PA = sin(120°)/6
To solve for side PA, we can rearrange the equation:
side PA = (side EA * sin(angle P))/sin(angle E)
Now we can substitute the known values:
side PA = (6 * sin(20°))/sin(120°)
Using a scientific calculator, we can find that sin(20°) is approximately 0.342 and sin(120°) is approximately 0.866. Plugging in these values, we get:
side PA = (6 * 0.342)/0.866
Evaluating this expression, we have:
side PA ≈ 2.38 inches
Now to find side PE, we can use the fact that the sum of the measures of the angles in a triangle is 180 degrees. Since we know two angles, we can find the third angle (angle A) by subtracting the sum of the other two angles from 180 degrees:
angle A = 180° - angle P - angle E
= 180° - 20° - 120°
= 40°
Now using the Law of Sines again, we can find side PE:
sin(angle E)/side PE = sin(angle A)/side EA
Plugging in the values we know, we get:
sin(120°)/side PE = sin(40°)/6
To solve for side PE, we can rearrange the equation:
side PE = (side EA * sin(angle E))/sin(angle A)
Now we can substitute the known values:
side PE = (6 * sin(120°))/sin(40°)
Using a scientific calculator again, we find that sin(120°) is approximately 0.866 and sin(40°) is approximately 0.642. Plugging in these values, we get:
side PE = (6 * 0.866)/0.642
Evaluating this expression:
side PE ≈ 8.07 inches
Therefore, the lengths of the other two sides of triangle PEA are approximately 2.38 inches and 8.07 inches.
Have you not learned the Sine Law.
It looks like a direct application of that.