Determine whether it is possible to draw a triangle given each set of information. Sketch all possible triangles where appropriate. Calculate then label all side lengths to the nearest tenth of a centimetre and all angles to the nearest degree.
A) b= 3.0 cm, c=5.5 cm and angle B = 30 degrees
First let's see if it is possible.
Start with a base of BC, sketch an angle of 30° at B and draw a line BA = 3.5
sketch in an altitude from A to BC to meet BC at D
sin30° = AD/3.5
AD = 3.5(1/2) = 1.75
Since AC > 1.75, it will be possible to draw the triangle. As a matter of fact there are two such triangles possible
(With centre at A and radius 3.0, we will be able to draw an arc which will cut BC at two places.
By the Sine Law:
sinC/3.5 = sin30/3
sinC = .58333..
angle C = 35.7° or 180-35.7 = 144.3°
Case1: C = 35.7 , then A = 114.3°
Case2: C = 144.3, then A = 5.7°
Use the sine law to find BC for each case.
This is called the ambigious case of the sine law, usually resulting in the data given as SSA
I misread your c = 5.5 as c = 3.5 , sorry about that
So AD should be 2.75
But AC is still greater than 2.75
so no real harm done, all you have to do is change the 3.5 to 5.5 and use the new values.
The conclusion is still the same, but make sure you get the new angles.
Well, well, well! We have a triangle problem here. Let's see if it's possible to draw a triangle with the given information.
First, we have side b = 3.0 cm and side c = 5.5 cm. To draw the triangle, the sum of any two sides must be greater than the remaining side.
So, let's add sides b and c. 3.0 + 5.5 = 8.5 cm.
And now, let's compare this sum with the remaining side, which is a. As long as a is greater than 8.5 cm, we're good to go!
But wait! We're not done yet. We also need to consider the angles. We have angle B = 30 degrees. The sum of the angles in a triangle is always 180 degrees.
The remaining angles (A and C) can be calculated using the Law of Sines. So let's find angle A first. We can use the sine rule:
sin(A) / a = sin(B) / b
sin(A) / a = sin(30) / 3.0
sin(A) = (sin(30) * a) / 3.0
Now, if we find the value of sin(A), we can calculate angle A.
Once we have angles A, B, and C, we can sketch the triangle and calculate the side lengths.
Alright, I'll leave the calculations to you. Remember to round the side lengths to the nearest tenth of a centimeter and the angles to the nearest degree.
And, don't forget, even if you can't draw a triangle, you can always draw a smiley face! Keep the humor alive!
To determine whether it is possible to draw a triangle given the information, we can use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.
For triangle ABC, with side lengths a, b, and c, and angles A, B, and C, the triangle inequality theorem can be written as:
a + b > c
b + c > a
c + a > b
In this case, we have side b = 3.0 cm and side c = 5.5 cm, and angle B = 30 degrees.
Using the triangle inequality theorem:
a + 3.0 cm > 5.5 cm
a > 5.5 cm - 3.0 cm
a > 2.5 cm
Therefore, side a must be greater than 2.5 cm for a triangle to be possible.
Now, let's sketch the triangle and calculate the missing angles and side lengths.
Step 1: Draw a line segment for side c = 5.5 cm.
Step 2: At one end of the line segment, draw an angle of 30 degrees. This will be angle B.
Step 3: From the other end of the line segment, draw a line segment of length 3.0 cm. This will be side b.
Step 4: Connect the loose ends of side b and side c to complete the triangle.
Step 5: Label angle B as 30 degrees.
Step 6: Label side b as 3.0 cm.
Step 7: Label side c as 5.5 cm.
Step 8: To calculate side a, we can use the law of cosines: a^2 = b^2 + c^2 - 2bc*cos(A)
a^2 = (3.0 cm)^2 + (5.5 cm)^2 - 2 * 3.0 cm * 5.5 cm * cos(30 degrees)
a^2 = 9 cm^2 + 30.25 cm^2 - 33 cm^2 * cos(30 degrees) = 39.25 cm^2 - 33 cm^2 * 0.866
a^2 = 39.25 cm^2 - 28.638 cm^2
a^2 = 10.612 cm^2
a ≈ 3.3 cm (rounded to the nearest tenth)
So, a triangle is possible, and the side lengths are approximately:
Side a ≈ 3.3 cm
Side b = 3.0 cm
Side c = 5.5 cm
Note: The accuracy of the angles and side lengths may vary depending on the precision of the calculations and the accuracy of the sketch.
To determine whether it is possible to draw a triangle given the set of information, we can use the triangle inequality theorem.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we are given:
b = 3.0 cm (length of side b)
c = 5.5 cm (length of side c)
angle B = 30 degrees
To use the triangle inequality theorem, we need to compare the sum of the lengths of two sides with the length of the third side.
Let's calculate the sum of the lengths of sides b and c:
b + c = 3.0 cm + 5.5 cm = 8.5 cm
Now let's compare this sum with the length of side a. We can use the formula for the triangle inequality theorem: a < b + c.
a < 8.5 cm
Since we don't have the length of side a given, we cannot determine if a triangle is possible with the given information.
However, if we assume that side a is the shortest side, we can calculate its length using the Law of Cosines.
The Law of Cosines states:
a^2 = b^2 + c^2 - 2bc * cos(A)
In this case, we are given:
b = 3.0 cm
c = 5.5 cm
angle B = 30 degrees
Let's calculate the length of side a:
a^2 = (3.0 cm)^2 + (5.5 cm)^2 - 2 * 3.0 cm * 5.5 cm * cos(30 degrees)
a^2 = 9.0 cm^2 + 30.25 cm^2 - 33.0 cm * cos(30 degrees)
a^2 = 9.0 cm^2 + 30.25 cm^2 - 33.0 cm * 0.866 (cosine of 30 degrees)
a^2 = 9.0 cm^2 + 30.25 cm^2 - 28.618 cm
a^2 = 10.632 cm^2
Taking the square root of both sides, we can find the length of side a:
a = √(10.632) cm
a ≈ 3.3 cm (rounded to the nearest tenth of a centimeter)
Now that we have the lengths of all three sides, we can sketch the triangle and label the side lengths:
Triangle ABC: A, B, and C represent the three vertices
Side a: BC
Side b: AC (length ≈ 3.0 cm)
Side c: AB (length ≈ 5.5 cm)
Please note that the triangle sketch would be best visualized on paper or using digital drawing tools.