Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and tan ∠X = 5 over 2 and 5 tenths.

Triangles XYZ and ACB; angles Y and C both measure 90 degrees, angles A and X are congruent.

Triangle XYZ was dilated by a scale factor of 2 to create triangle ACB and tan ∠X = 5 over 2 and 5 tenths.

Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles XYZ and ACB. You must show all work and calculations to receive full credit. (5 points)

Part B: Explain how to find the measures of segments AC and CB. You must show all work and calculations to receive full credit. (5 points)

According to the Angle-Angle Similarity, if any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.

Therefore, the sides of ΔXYZ and ΔACB are in the same ratio.

Tan trigonometric ratio

[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]

where:

θ is the angle.
O is the side opposite the angle.
A is the side adjacent the angle.

Therefore, if:

[tex]\sf \tan \angle X=\dfrac{5}{2.5}[/tex]

and ΔACB is a dilation of ΔXYZ by scale factor 2:

[tex]\implies \sf \tan \angle A=\dfrac{2 \cdot 5}{2 \cdot 2.5}=\dfrac{5}{2.5}[/tex]

So the trigonometric ratios of ΔACB and ΔXYZ are the same.

Part B

The side opposite angle X is YZ and the side adjacent angle X is XY.

Using the tan trigonometric ratio and the given value of tan X:

[tex]\implies \sf \tan \angle X=\dfrac{O}{A}=\dfrac{YZ}{XY}=\dfrac{5}{2.5}[/tex]

Therefore:

YZ = 5
XY = 2.5

As ΔACB ~ ΔXYZ and ΔACB is a dilation of ΔXYZ by scale factor 2:

AC = 2(XY)
CB = 2(YZ)

Therefore:

⇒ AC = 2(2.5) = 5

⇒ CB = 2(5) = 10