Question

connecting concepts the larger triangle is a dilation of the smaller triangle. find the values of x and y (3y - 34)° 2x + 8 (y + 16)° x + 1 c 2 6

Explanation:

Step1: Find the scale factor of dilation

The ratio of corresponding sides should be equal for similar triangles (since dilation preserves similarity). The base of the smaller triangle is 2 and the base of the larger triangle is 6, so the scale factor \( k=\frac{6}{2} = 3 \).

Step2: Solve for \( x \) using the scale factor

The side of the smaller triangle is \( x + 1 \) and the corresponding side of the larger triangle is \( 2x + 8 \). Since the scale factor is 3, we have \( 2x + 8=3(x + 1) \).
Expand the right - hand side: \( 2x+8 = 3x+3 \).
Subtract \( 2x \) from both sides: \( 8=x + 3 \).
Subtract 3 from both sides: \( x=8 - 3=5 \).

Step3: Solve for \( y \) using the scale factor of angles (since dilation preserves angle measures)

The angle of the smaller triangle is \( (y + 16)^{\circ} \) and the corresponding angle of the larger triangle is \( (3y-34)^{\circ} \). So we set up the equation \( 3y-34=y + 16 \).
Subtract \( y \) from both sides: \( 2y-34 = 16 \).
Add 34 to both sides: \( 2y=16 + 34=50 \).
Divide both sides by 2: \( y=\frac{50}{2}=25 \).

Answer:

\( x = 5 \), \( y=25 \)