Respuesta :
Geometric mean (leg) theorem states that the altitude drawn form the hypotenuse side of the right angle triangle divides the hypotenuse side into two equal sides and the length of the leg of this triangle is geometric mean between hypotenuse and segment of the hypotenuse adjacent to that leg. The value of [tex]a[/tex] is [tex]2\sqrt{70}[/tex] units. Thus the option B is the correct option.
Given information-
Length of side AD is 6 units.
Length of side BD is 14 units.
Length of side BC is a units.
Length of the altitude AD is h units.
Geometric mean theorem
Geometric mean (leg) theorem states that the altitude drawn form the hypotenuse side of the right angle triangle divides the hypotenuse side into two equal sides and the length of the leg of this triangle is geometric mean between hypotenuse and segment of the hypotenuse adjacent to that leg.
In the given [tex]\Delta ABC[/tex] the [tex]\angle C[/tex] is the right angle triangle. AD is the altitude drawn from the right angle to the hypotenuse of the [tex]\Delta ABC[/tex]. Thus by the geometric mean theorem,
[tex]\dfrac{h}{6} =\dfrac{14}{h}\\h^2=14\times6\\h=\sqrt{84} \\h=2\sqrt{21}[/tex]
Thus the length of the altitude h is [tex]2\sqrt{21}[/tex] units.
Now in right angle [tex]\Delta BDC[/tex] hypotenuse is [tex]a[/tex]. Thus by the Pythagoras theorem,
[tex]a^2=14^2+h^2[/tex]
Put the values,
[tex]a^2=14^2+(2\sqrt{21}) ^2\\a^2=196+84\\[/tex]
Solve for a,
[tex]a=\sqrt{196+84} \\a=\sqrt{280}\\ a=2\sqrt{70}[/tex]
Hence the value of [tex]a[/tex] is [tex]2\sqrt{70}[/tex] units. Thus the option B is the correct option.
Learn more about the geometric mean theorem here;
https://brainly.com/question/10612854
Answer:
B 2 square 70
Step-by-step explanation:
hope this helps and good luck!!
