Prove the triangle proportionality theorem.

Statement of triangle proportionality:  

If a line parallel to one side of a triangle intersects the other two sides of the triangle, then that line divides these two sides proportionally.

From the statement: If then,

Show that:

Consider and

Reflexive property states that the value is equal to itself.

[Angle] {Reflexive property of equality}

Corresponding angles theorem states that if the two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent(i., e equal).

[Angle]

[Angle]

AA Similarity states that the two triangles have their corresponding angles equal if and only if their corresponding sides are proportional.

then, by AA similarity theorem:  

By segment addition postulates:

AB = FA +FB and AC = AG + GC  

Corresponding sides in similar triangles are proportional  

.....[1]

Substitute AB = FA +FB and AC = AG + GC in [1]  

we have;

Separate the fraction:

Simplify:

Subtract 1 from both sides we get;

hence proved

let f (x ) is a function and c be a positive real number,   the graph of the function g (x ) = f (cx ) is represented as follows.

when, c > 1,   the function compresses it in the x - direction.

when, 0 < c < 1, the function stretches it.

the function is g ( x ) = (9x )2.

the heighest power of x in g ( x ) = (9x )2 is 2, so the parent function is square function.

the equation of parent function is f ( x ) = x 2 .

the function is g ( x ) = (9x )2 and the parent function is f ( x ) = x 2 .

so, the functions f and g have the following relationship.

g ( x ) = f ( 9x ).

compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

so, the function g (x ) = (9x )2 compresses it in the x - direction.

step-by-step explanation:

step by step explanation: when u removing the parentheses and combined them that would be : x+3y-x-y=2y