Question

calculator
which pairs of rectangles are similar polygons?
select similar or not similar for each pair of rectangles.
similar not similar
a and b

b and c

a and c

120 cm
b
100 cm
100 cm
a
80 cm
90 cm
c
72 cm

Explanation:

Step1: Recall Similar Rectangle Condition

For rectangles to be similar, their corresponding side ratios must be equal. So, we check the ratio of length to width (or width to length) for each pair.

Step2: Check Pair A and B

Rectangle A: dimensions \(100\,\text{cm}\) (height) and \(80\,\text{cm}\) (width). Ratio of height to width: \(\frac{100}{80}=\frac{5}{4} = 1.25\).
Rectangle B: dimensions \(120\,\text{cm}\) (height) and \(100\,\text{cm}\) (width). Ratio of height to width: \(\frac{120}{100}=\frac{6}{5}=1.2\).
Since \(\frac{5}{4}
eq\frac{6}{5}\), A and B are Not Similar.

Step3: Check Pair B and C

Rectangle B: ratio \(\frac{120}{100}=\frac{6}{5} = 1.2\) (height to width) or \(\frac{100}{120}=\frac{5}{6}\) (width to height).
Rectangle C: dimensions \(90\,\text{cm}\) (height) and \(72\,\text{cm}\) (width). Ratio of height to width: \(\frac{90}{72}=\frac{5}{4}=1.25\), or width to height: \(\frac{72}{90}=\frac{4}{5}=0.8\). Wait, maybe check width to height for B: \(\frac{100}{120}=\frac{5}{6}\approx0.833\), and for C: \(\frac{72}{90}=\frac{4}{5}=0.8\). Not equal. Wait, maybe height to width for B: \(120/100 = 6/5\), for C: \(90/72 = 5/4\). Not equal. Wait, maybe I mixed up. Wait, rectangle A: 100 (height) and 80 (width) → ratio 100/80 = 5/4. Rectangle C: 90 (height) and 72 (width) → 90/72 = 5/4. Oh! Wait, maybe I should check width to height for B. Wait, let's re-express:

Wait, maybe I took height and width incorrectly. Let's define length and width as the two sides. For rectangles, similar means corresponding sides are proportional. So, for A: sides 100 and 80. For B: sides 120 and 100. For C: sides 90 and 72.

Check A and C: A's sides: 100, 80. C's sides: 90, 72. Let's see if \( \frac{100}{90}=\frac{80}{72} \)? Simplify: \( \frac{10}{9} \) vs \( \frac{10}{9} \) (since \( \frac{80}{72}=\frac{10}{9} \), \( \frac{100}{90}=\frac{10}{9} \)). Wait, wait, earlier mistake! Let's redo:

Wait, maybe I assigned height and width wrong. Let's take the longer side as length. For A: 100 (length) and 80 (width). For B: 120 (length) and 100 (width). For C: 90 (length) and 72 (width).

Now, check A and B: ratio of length to width: A: \(100/80 = 5/4 = 1.25\); B: \(120/100 = 6/5 = 1.2\). Not equal. So A and B: Not Similar.

Check B and C: B: \(120/100 = 6/5 = 1.2\); C: \(90/72 = 5/4 = 1.25\). Not equal. So B and C: Not Similar? Wait, no, wait A and C: A: \(100/80 = 5/4\); C: \(90/72 = 5/4\). Oh! I see, I messed up B and C. Let's correct:

Rectangle A: length 100, width 80 → ratio \(100/80 = 5/4\).
Rectangle C: length 90, width 72 → ratio \(90/72 = 5/4\). So A and C have the same ratio.

Wait, let's recheck B: length 120, width 100 → ratio \(120/100 = 6/5\). So:

• A and B: ratios \(5/4\) vs \(6/5\) → Not Similar.
• B and C: ratios \(6/5\) vs \(5/4\) → Not Similar.
• A and C: ratios \(5/4\) vs \(5/4\) → Similar.

Wait, I think I made a mistake in Step 3 earlier. Let's redo:

Step2 (corrected) Check A and B:

A: \(100/80 = 5/4\); B: \(120/100 = 6/5\). Not equal → Not Similar.

Step3 Check B and C:

B: \(120/100 = 6/5\); C: \(90/72 = 5/4\). Not equal → Not Similar.

Step4 Check A and C:

A: \(100/80 = 5/4\); C: \(90/72 = 5/4\). Equal → Similar.

So:

• A and B: Not Similar
• B and C: Not Similar
• A and C: Similar

Answer:

• A and B: Not Similar
• B and C: Not Similar
• A and C: Similar

(In the table, for A and B: select Not Similar; for B and C: select Not Similar; for A and C: select Similar)