In triangle ABC, D is the midpoint of BC. E is the foot of the perpendicular from C to AD. If CD=7√,DE=3√, what is the measure (in degrees) of ∠BED?
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To find the measure of ∠BED, we need to use trigonometry and apply the Pythagorean theorem.
Let's start by drawing triangle ABC and point D as the midpoint of BC. Then, draw point E as the foot of the perpendicular from C to AD.
Since CD is given as 7√, we can label CD as 7√. Additionally, DE is given as 3√, so we can label DE as 3√.
Now, using the Pythagorean theorem, we can find the length of AD. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, AD) is equal to the sum of the squares of the other two sides (CD and DE).
So, we have:
AD^2 = CD^2 + DE^2
AD^2 = (7√)^2 + (3√)^2
AD^2 = 49 + 9
AD^2 = 58
To simplify, take the square root of both sides:
AD = √58
Now, let's focus on triangle ADE. We know the lengths of AD and DE, and we need to find the measure of ∠BED.
To find the measure of ∠BED, we can use the sine function. The sine of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, we want to find sin(∠BED).
sin(∠BED) = DE / AD
sin(∠BED) = (3√) / (√58)
To find the measure of ∠BED, we need to find the inverse sine (also known as arcsine) of sin(∠BED):
∠BED = arcsin(de / AD)
∠BED = arcsin((3√) / (√58))
To obtain the measure in degrees, use a calculator to find the arcsine of ((3√) / (√58)) and convert the result from radians to degrees.
So, the final step is to substitute the value of ((3√) / (√58)) into a calculator to find the arcsine and convert the result into degrees to find the measure of ∠BED.