Question

question 1-39
consider rectangles abcd and efgh as shown in the graph.
which statement is true?
○ $ab : ef = 3 : 1$
○ $ab : ef = 2 : 3$
○ perimeter of abcd : perimeter of efgh = $1 : 3$
○ efgh is a dilation of abcd with a scale factor of $\frac{1}{3}$.

Explanation:

Step1: Determine length of AB

From the graph, AB is vertical. Let's count the grid units. A is at y=3, B at y=1, so length AB = 3 - 1 = 2 units.

Step2: Determine length of EF

EF is vertical. E is at y=9, F at y=3, so length EF = 9 - 3 = 6 units.

Step3: Analyze AB:EF ratio

AB = 2, EF = 6, so AB:EF = 2:6 = 1:3. Wait, but let's check perimeters.

Step4: Perimeter of ABCD

Let's find dimensions of ABCD. AB=2, AD: from x=-9 to x=-6, so AD=3 units. Perimeter of rectangle: 2(length + width) = 2(2 + 3) = 10? Wait, no, wait. Wait, ABCD: A(-9,3), B(-9,1), C(-6,1), D(-6,3). So length AB (vertical) is 3 - 1 = 2, length BC (horizontal) is -6 - (-9) = 3. So perimeter of ABCD: 2(2 + 3) = 10? Wait, no, 2(length + width) = 2(3 + 2) = 10? Wait, horizontal side: from x=-9 to x=-6, that's 3 units (width), vertical side: 2 units (height). So perimeter: 2(3 + 2) = 10.

Step5: Perimeter of EFGH

E(-1,9), F(-1,3), G(8,3), H(8,9). So vertical side EF: 9 - 3 = 6, horizontal side FG: 8 - (-1) = 9. Perimeter of EFGH: 2*(9 + 6) = 30.

Step6: Analyze perimeter ratio

Perimeter of ABCD : Perimeter of EFGH = 10 : 30 = 1 : 3.

Now check options:

• AB:EF = 2:6 = 1:3, so first option (3:1) wrong, second (2:3) wrong.

• Perimeter ratio: 10:30 = 1:3, so third option is correct.

• Dilation: EFGH is larger than ABCD, so scale factor should be 3 (since EF=6, AB=2, 6/2=3), so fourth option (scale factor 1/3) is wrong.

Answer:

Perimeter of \( ABCD \) : Perimeter of \( EFGH = 1 : 3 \) (the third option)