The figure shows three right triangles. triangles uvw, uzv, and vzw are similar. theorem: if two triangles are similar, the corresponding sides are in proportion. the figure shows triangle uvw with right angle at v. segment uv is 2 and segment vw is 3. point z is on segment uw and angles vzu and vzw are right angles. using the given theorem, which two statements to prove that if segment uw is x, then x2 = 13? segment uw • segment uz = 4 segment uw • segment zw = 6 segment uw • segment uz = 9 segment uw • segment zw = 4 segment uw • segment uz = 4 segment uw • segment vw = 9 segment uw • segment uz = 6 segment uw • segment zw = 4

We can see the figure of triangles in attachment as shown

In triangles UVW and VZW

∠UVW = ∠VZW (both are equal to 90°)

∠VWU = ∠VWZ ( common angle W in both triangles)

So by AA property Δ UVW ~ ΔVZW

So its coressponding sides will have equal ratio

So

Now plugging 3 in VW place we get

Now on cross multiplying we get

UV × ZW = 3 × 3

UV × ZW = 9

so clearly all choices (1),(2) and (4) are incorrect as they state

UV × ZW = 4

As Δ UVW ~ ΔVZW and so corresponding sides will have equal ratios so we can also have

Now if you plug 3 in VW place here we get

Now cross multiply

UW × ZW = 3 × 3

UW × ZW = 9 ----------------------(1)

Please recheck choice (3) second statement must have been

Segment UW • segment ZW = 9, instead of VW its must be ZW

So now lets check choice (3) first statement

For that we can use triangles UVW and UZV

Here ∠UVW = ∠UZV (both are equal to 90°)

∠VUW = ∠VUZ ( common angle U in both triangles)

So by AA property Δ UVW ~ ΔUZV

So its coressponding sides will have equal ratio

So

now plug 2 in UV place here

now cross multiply

UW × UZ = 2 × 2

UW × UZ = 4 -------------------(2)

which is there in choice (3) first statement

Segment UW • segment UZ = 4

From equation (1)

UW × ZW = 9

plug x in UW place and solve for ZW

x × ZW = 9-----------------------------------(3)

similarly plug x in UW place in equation (2) also so we get

x × UZ = 4 -----------------------------(4)

Now add equations (3) and (4)

x × ZW + x × UZ = 9 + 4

x × ZW + x × UZ = 13

x is common in both terms so can take x common out

x (ZW +UZ) = 13

now ZW+UZ = UW, can see in figure in attachment

so x(UW) = 13

again plug x in UW place

so x(x) =13

hence is proved

so choice (3) is the right answer

Segment UW • segment UZ = 4

Segment UW • segment ZW = 9

they used 313 gallons of paint

the 8 in 8,420 is in the thousands place so it's value is 8 thousand.

the 8 in 6,870 is in the hundreds place so it's value is 8 hundred.

divide 8000 by 800 to find the answer:

8000 / 800 = 10

the 8 in 8,420 is 10 times greater than the 8 in 6,870