Triangle MNO is similar to triangle RPO.

Triangle M N O. Side M N is 20 kilometers and side M O is 32 kilometers. Triangle R P O. Side O R is 48 kilometers.

Valek finds the distance between P and R. His work is shown below.

Step 1 StartFraction 32 Over 48 EndFraction = StartFraction 20 Over P R EndFraction

Step 2 20 P R = (32) (48)

Step 3 20 P R = 1,536

Step 4 P R = 76.8 kilometers

What is Valek’s first error?
Valek did not correctly divide 1,536 by 20 going from step 3 to step 4.

Valek did not find the correct product of 32 and 48 going from step 2 to step 3.

Valek should have written the proportion in step 1 as StartFraction 32 Over 20 EndFraction = StartFraction P R Over 48 EndFraction.

Valek should have written the cross-product in step 2 as 32 P R = (20) (48).

Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles. Similar triangles are the triangles that are the same in shape, but may not be equal in size.

For the given situation,

Triangle MNO is similar to triangle RPO.

For triangle MNO, the sides are MN = 20 kilometers and MO = 32 kilometers.

For triangle RPO, The side is OR = 48 kilometers.

If two triangles are similar, then the ratio of their corresponding sides are equal.

[tex]\frac{MO}{OR}=\frac{MN}{PR}[/tex]

⇒ [tex]\frac{32}{48} =\frac{20}{PR}[/tex]

⇒ [tex]32PR=(20)(48)[/tex]

⇒ [tex]PR=\frac{(20)(48)}{32}[/tex]

⇒ [tex]PR=\frac{960}{32}[/tex]

⇒ [tex]PR=30[/tex]

These are the correct steps to find the side PR.

Valek did the mistake in step 2.

Hence we can conclude that Valek's first error is in step 2. The correct answer is PR = 30 kilometers.

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