PLEASE HELP
In the triangle, the length of side b is 7 the m∠A=60°. Find the exact lengths of a and c. Exact means you CANNOT give me decimals. You will need to give me either an integer a fraction or a radical. Use the unit circle.
You have given only two pieces of information about the triangle,
so the measure of the other sides and angles cannot be found.
the bot's answer is meaningless
Just about the only way to get an exact answer is to use special angles whose trig functions are known. If m∠A=60° and m∠B=30°, then
if b=7, a=7√3 and c=14
To find the lengths of sides a and c in the triangle, we can use the trigonometric ratios in combination with the given information.
First, let's analyze the given information and what it tells us about the triangle:
- Length of side b = 7
- Measure of angle A = 60°
We can assign labels to the sides of the triangle corresponding to the given information:
- Side a is opposite angle A.
- Side b is the side whose length is given as 7.
- Side c is the remaining side.
To find the length of side a, we can use the sine function (sin).
The sine function relates the length of the side opposite an angle to the length of the hypotenuse (the longest side in a right triangle).
In this case, since triangle ABC is not a right triangle, we need to create a right triangle from it.
To do this, we draw an altitude from angle A to side b, splitting it into two shorter segments.
Let's label the point where the altitude intersects side b as point D.
Now, we have a right triangle ACD, with angle C as 90°.
We know the following:
- Length of side b = AD + DB = 7 (given)
- Angle A = 60° (given)
Using the right triangle ACD, we can find the value of sin(60°) by considering the ratio of the length of side opposite angle A (side a) to the length of the hypotenuse (side c).
sin(A) = opposite/hypotenuse = a/c
Now, let's apply the unit circle concept to find the exact value of sin(60°):
On the unit circle, the angle measuring 60° is associated with the point (√3/2, 1/2).
This means that sin(60°) = y-coordinate of the point on the unit circle = 1/2.
Substituting this value into the trigonometric ratio equation, we have:
1/2 = a/c
Solving for a, we get:
a = c/2
Now, let's go back to our right triangle ACD.
Since we know the length of side b is 7, using the Pythagorean theorem, we can find the value of side c.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In triangle ACD, we have:
AD^2 + CD^2 = AC^2
Since AD = a and CD = b, we can substitute these values:
a^2 + 7^2 = c^2
a^2 + 49 = c^2
Since a = c/2, we can substitute this value:
(c/2)^2 + 49 = c^2
(c^2/4) + 49 = c^2
Multiplying through by 4 to get rid of the fraction:
c^2 + 196 = 4c^2
3c^2 = 196
Simplifying, we find:
c^2 = 196/3
Taking the square root of both sides, we get:
c = √(196/3)
Hence, the exact length of side c is √(196/3).
Finally, we can substitute the value of side c into the equation a = c/2 to find the length of side a:
a = (√(196/3))/2
Therefore, the exact length of side a is (√(196/3))/2.
a = 6√3
c = 6