Question

diana is creating the graphics for a series of shirts. she initially designed the graphics for a medium shirt with the dimensions shown below.
the company printing the shirts needs the adjusted dimensions of the graphics for the small and large shirts. diana intends to provide measurements that ensure the graphics on all shirts are similar in shape, no matter the size.
which of the following statements is true?
a. if ih on the large shirt is 12 cm, then bc is 2.5 cm.
b. if gh on the small shirt is 3.5 cm, then de is 1.25 cm.
c. if fe on the small shirt is 1.5 cm, then ih is 7.25 cm.
d. if ab on the large shirt is 3.5 cm, then gi is 12.25 cm.

Explanation:

Step1: Identify Similar Figures

The original graphics have similar triangles. For similar figures, the ratios of corresponding sides are equal. Let's find the scale factor for each option.

Step2: Analyze Option A

Original \( IH = 10 \, \text{cm} \), \( BC = 2 \, \text{cm} \). New \( I'H' = 12 \, \text{cm} \). Scale factor \( k = \frac{12}{10} = 1.2 \). Then \( B'C' = 2 \times 1.2 = 2.4
eq 2.5 \). So A is false.

Step3: Analyze Option B

Original \( GH = 7 \, \text{cm} \) (wait, no, original \( DE = 2 \, \text{cm} \), \( GH \) is part of the isosceles triangle? Wait, original \( FE = 2 \, \text{cm} \), \( DE = 2 \, \text{cm} \), \( IH = 10 \, \text{cm} \), \( GI = 7 \, \text{cm} \), \( BC = 2 \, \text{cm} \), \( AB = 2 \, \text{cm} \). For the small shirt, \( G'H' = 3.5 \, \text{cm} \). Original \( GI = 7 \, \text{cm} \), so scale factor \( k = \frac{3.5}{7} = 0.5 \). Then \( D'E' = 2 \times 0.5 = 1
eq 1.25 \). So B is false.

Step4: Analyze Option C

Original \( FE = 2 \, \text{cm} \), new \( F'E' = 1.5 \, \text{cm} \). Scale factor \( k = \frac{1.5}{2} = 0.75 \). Then \( I'H' = 10 \times 0.75 = 7.5
eq 7.25 \). So C is false.

Step5: Analyze Option D

Original \( AB = 2 \, \text{cm} \), new \( A'B' = 3.5 \, \text{cm} \). Scale factor \( k = \frac{3.5}{2} = 1.75 \). Original \( GI = 7 \, \text{cm} \), so \( G'I' = 7 \times 1.75 = 12.25 \, \text{cm} \). So D is true.

Answer:

D. If \( A'B' \) on the large shirt is 3.5 cm, then \( G'I' \) is 12.25 cm.