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

Question

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

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)

Answer 1

Part A

Triangles XYZ and ACB exhibit a special relationship due to the dilation by a scale factor of 2. When triangle XYZ is dilated to form triangle ACB, all corresponding side lengths in triangle ACB are twice as large compared to triangle XYZ. This is a key feature of geometric dilations: although the triangles are different sizes, their angles remain congruent, and their shape stays the same.

Given that \( \tan ∠X = \frac{5}{2.5} = 2 \), we can infer specific ratios between the lengths of the sides adjacent and opposite to angle X in triangle XYZ.

In triangle XYZ, if we let the length of the side opposite to \( \angle X \) (let’s call it segment \( YZ \)) be 2 units, then the length of the side adjacent to \( \angle X \) (segment \( XY \)) would be 5 units, based on the tangent ratio: \[ \tan ∠X = \frac{\text{opposite}}{\text{adjacent}} = \frac{YZ}{XY} = \frac{2}{5}. \]

Since triangle ACB is a scaled-up version of triangle XYZ (scale factor of 2), the lengths of the opposite and adjacent sides in triangle ACB (let's denote them as \( AC \) and \( CB \) respectively) would be: \[ AC = 2 \cdot YZ = 2 \cdot 2 = 4 \text{ units}, \] \[ CB = 2 \cdot XY = 2 \cdot 5 = 10 \text{ units}. \]

Thus, the trigonometric ratios maintained their consistency in both triangles, demonstrating that the properties of similar triangles extend to include the relationships between their angles and side lengths.

Part B

To find the measures of segments AC and CB in triangle ACB, we need to leverage the results from part A and the properties of similar triangles.

Since we established that triangle XYZ was dilated by a scale factor of 2, we apply this factor to the corresponding sides of triangle XYZ:

1. Finding segment AC:

   • In triangle XYZ, the length of segment \( YZ \) (the side opposite angle X) is 2 units.
   • Applying the scale factor of 2 to find corresponding length in triangle ACB gives: \[ AC = 2 \cdot YZ = 2 \cdot 2 = 4 \text{ units}. \]

2. Finding segment CB:

   • In triangle XYZ, the length of segment \( XY \) (the side adjacent to angle X) is 5 units.
   • Again, applying the scale factor of 2 gives: \[ CB = 2 \cdot XY = 2 \cdot 5 = 10 \text{ units}. \]

In summary, after performing the calculations:

• The measure of segment AC is 4 units.
• The measure of segment CB is 10 units.

Thus, we find: \[ AC = 4 \text{ units} \quad \text{and} \quad CB = 10 \text{ units}. \]