The following statements describe triangles ABC and PQR.

Question

For ΔABC
: AC=2, AB=4, and BC=5

For ΔPQR
: QR=7.5, PR=3, and PQ=6

Which statement explains why ΔABC
 and ΔPQR
 are either similar or not similar.

(6 points)
Responses

ΔABC
 and ΔPQR
 are not similar because  ACQR≠ABPR
.
   ΔABC
 and ΔPQR
 are not similar because       cap A cap c over cap q cap r is not equal to cap A cap b over cap p cap r.

ΔABC
 and ΔPQR
 are similar because  ACPR=PQAB=BCQR
. 
   ΔABC
 and ΔPQR
 are similar because     cap A cap c over cap p cap r is equal to cap p cap q over cap A cap b is equal to cap b cap c over cap q cap r.

ΔABC
 and ΔPQR
 are similar because ABPQ=BCQR
   ΔABC
 and ΔPQR
 are similar because    cap A cap b over cap p cap q is equal to cap b cap c over cap q cap r

ΔABC
 and ΔPQR
 are similar because ABPR=BCQR=ACPQ
   ΔABC
 and ΔPQR
 are similar   because    cap A cap b over cap p cap r is equal to cap b cap c over cap q cap r is equal to cap A cap c over cap p cap q

GPT-4o mini · Answer

To determine whether triangles ABC and PQR are similar, we can use the Side-Side-Side (SSS) similarity criterion, which states that two triangles are similar if the lengths of their corresponding sides are proportional.

Let’s examine the sides given:

For ΔABC:

AC = 2
AB = 4
BC = 5

For ΔPQR:

QR = 7.5
PR = 3
PQ = 6

Now, we can find the ratios of the corresponding sides:

Ratio of AB to PQ: \[ \frac{AB}{PQ} = \frac{4}{6} = \frac{2}{3} \]
Ratio of AC to PR: \[ \frac{AC}{PR} = \frac{2}{3} \]
Ratio of BC to QR: \[ \frac{BC}{QR} = \frac{5}{7.5} = \frac{5}{7.5} = \frac{2}{3} \]

Since all corresponding sides have the same ratio of \(\frac{2}{3}\), triangles ABC and PQR are similar by SSS.

Based on the provided response options, the correct statement that articulates similarity due to proportional sides is:

\[ \Delta ABC \text{ and } \Delta PQR \text{ are similar because } \frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR}. \]

Thus, the option that states: ΔABC and ΔPQR are similar because \(\frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR}\) would explain their similarity correctly.