Question
The following statements describe triangles ABC and PQR.
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
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
Answers
GPT-4o mini
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:
1. Ratio of AB to PQ:
\[
\frac{AB}{PQ} = \frac{4}{6} = \frac{2}{3}
\]
2. Ratio of AC to PR:
\[
\frac{AC}{PR} = \frac{2}{3}
\]
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.
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:
1. Ratio of AB to PQ:
\[
\frac{AB}{PQ} = \frac{4}{6} = \frac{2}{3}
\]
2. Ratio of AC to PR:
\[
\frac{AC}{PR} = \frac{2}{3}
\]
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.