F(x) = x + 7, g(x) = x - 7 (a) (f + g)(x) = (b) (f-g)(x)= (c) (fg)(x)= (d) (f/g)(x)=

In right △ABC, the altitude CH to the hypotenuse AB intersects angle bisector AL in point D. Find the sides of △ABC if AD = 8 cm

tangare [24]

Answer:

$AC=8\sqrt{3}\ cm\\ \\AB=16\sqrt{3}\ cm\\ \\BC=24\ cm$

Step-by-step explanation:

Consider right triangle ADH ( it is right triangle, because CH is the altitude). In this triangle, the hypotenuse AD = 8 cm and the leg DH = 4 cm. If the leg is half of the hypotenuse, then the opposite to this leg angle is equal to 30°.

By the Pythagorean theorem,

$AD^2=AH^2+DH^2\\ \\8^2=AH^2+4^2\\ \\AH^2=64-16=48\\ \\AH=\sqrt{48}=4\sqrt{3}\ cm$

AL is angle A bisector, then angle A is 60°. Use the angle's bisector property:

$\dfrac{CA}{CD}=\dfrac{AH}{HD}\\ \\\dfrac{CA}{CD}=\dfrac{4\sqrt{3}}{4}=\sqrt{3}\Rightarrow CA=\sqrt{3}CD$

Consider right triangle CAH.By the Pythagorean theorem,

$CA^2=CH^2+AH^2\\ \\(\sqrt{3}CD)^2=(CD+4)^2+(4\sqrt{3})^2\\ \\3CD^2=CD^2+8CD+16+48\\ \\2CD^2-8CD-64=0\\ \\CD^2-4CD-32=0\\ \\D=(-4)^2-4\cdot 1\cdot (-32)=16+128=144\\ \\CD_{1,2}=\dfrac{-(-4)\pm\sqrt{144}}{2\cdot 1}=\dfrac{4\pm 12}{2}=-4,\ 8$

The length cannot be negative, so CD=8 cm and

$CA=\sqrt{3}CD=8\sqrt{3}\ cm$

In right triangle ABC, angle B = 90° - 60° = 30°, leg AC is opposite to 30°, and the hypotenuse AB is twice the leg AC. Hence,

$AB=2CA=16\sqrt{3}\ cm$

By the Pythagorean theorem,

$BC^2=AB^2-AC^2\\ \\BC^2=(16\sqrt{3})^2-(8\sqrt{3})^2=256\cdot 3-64\cdot 3=576\\ \\BC=24\ cm$

3 0

2 years ago

A group of mathematical symbols that express a relationship or that are used to solve a problem

pogonyaev

A group of mathematical symbols that express a relationship or that are used to solve a problem is a FORMULA

8 0

2 years ago

How do you draw a skew line?

Bas_tet [7]

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. Hope this helps!! :)

5 0

3 years ago

can anyone help?????? Can y’all write it detailed or like step by step explaining cause i have to write it down ..

inysia [295]

Answer:

x = 5

Step-by-step explanation:

Step 1: Distributive property

Using this property means when you have something like the equation on the right side ( 4 ( x + 3 ) ) you multiply both the values in the parentheses by the number outside:

4 ( x + 3 )

( 4 x X ) + ( 4 x 3 )

4x + 12

8x + 2 = 2x + 4x + 12

Step 2: combine like terms

Combining like terms is when you find like terms with the same variables and such and add them together:

8x + 2 = 12 + ( 2x + 4x )

8x + 2 = 12 + 6x

Step 3:Using inverse operations

This means that you need to get a variable one one side and a constant on the other:

8x + 2 ( -2 ) = 12 ( -2 ) + 6x

8x = 10 + 6x

8x ( -6x ) = 10 + 6x ( -6x )

2x = 10

Step 4: solve for the variable

The last thing you need to do is divide both sides by the constant with the variable ( 2x ) to get x by itself:

2x ( /2 ) = 10 ( /2 )

x = 5

7 0

2 years ago

Read 2 more answers

Jack leaves from school and goes 4 miles north and 3 miles east to reach his home. How far is his home from school?

fiasKO [112]

Answer:

D. 5 miles

Step-by-step explanation:

Jack forms a right triangle with his route: the distance between the school and home is the hypotenuse.

It is equal to:

√4²+3²= √25 = 5 miles

Correct choice is D

3 0

2 years ago