Triangle Similarity Test
To determine if \(\Delta\text{ABC}\) and \(\Delta\text{PQR}\) are similar, we check if their corresponding sides are in proportion using the SSS Similarity criterion.
Given side lengths:
- \(\Delta\text{ABC}\): \(\text{AB} = 2\text{cm}\), \(\text{BC} = 4\text{cm}\), \(\text{AC} = 5\text{ cm}\)
- \(\Delta\text{PQR}\): \(\text{PQ} = 12\text{cm}\), \(\text{QR} = 24\text{ cm}\), \(\text{PR} = 30\text{ cm}\)
Proportional Sides Check
Compare the ratios of corresponding sides. The correspondence is inferred from the order: A \(\leftrightarrow\) P, B \(\leftrightarrow\) Q, C \(\leftrightarrow\) R.
- Ratio of sides PQ and AB: \(\frac{\text{PQ}}{\text{AB}} = \frac{12}{2} = 6\)
- Ratio of sides QR and BC: \(\frac{\text{QR}}{\text{BC}} = \frac{24}{4} = 6\)
- Ratio of sides PR and AC: \(\frac{\text{PR}}{\text{AC}} = \frac{30}{5} = 6\)
All three ratios are equal to 6. This indicates that the sides are proportional.
Triangle Relationship Conclusion
Since the ratios of all corresponding sides are equal (\(\frac{\text{PQ}}{\text{AB}} = \frac{\text{QR}}{\text{BC}} = \frac{\text{PR}}{\text{AC}} = 6\)), \(\Delta\text{ABC}\) and \(\Delta\text{PQR}\) are Similar by the SSS Similarity criterion.
Similar Questions
- The areas of two similar triangles $\Delta$XYZ and $\Delta$LMN are 49 cm$^2$ and 9 cm$^2$, respectively. If LM = 9 cm, then the length of XY is:
Triangles \(PQR\) and \(ABC\) are similar triangles. Triangle \(ABC\) have the following sides
\(AB = 12 \text{ cm}\),
\(AC = 15 \text{ cm}\),
\(CB = 21 \text{ cm}\).
Triangle \(PQR\) have the following sides:
\(PQ = 4 \text{ cm}\),
\(RQ = 7 \text{ cm}\).
What is the length of side \(PR\)?
For a pair of similar triangles, the angles made at \(Q\) is same in both the triangles \(PQR\) and \(LQJ\).
If the length of side \(QJ\) is 10 cm and \(QR\) is 5 cm, what is the ratio of their total areas?- \(\Delta OPQ\) is similar to \(\Delta RST\). If the ratio of OP : RS is 3 : 5 and if PQ = 6 cm, then the length of ST is:
- Which of the following is the property of similar triangle?
- Two triangle are called similar if
- The areas of two similar triangles \(\Delta XYZ\) and \(\Delta LMN\) are \(49\ cm^2\) and \(9\ cm^2\), respectively. If LM = 9 cm, then the length of XY is:
Important Questions from Triangles, Congruence and Similarity
In a $\Delta ABC$, $\angle C = (9/4) \angle B = (3/2) (\angle A+\angle B)$. Find the measure of the angles.
- The areas of two similar triangles $\Delta$XYZ and $\Delta$LMN are 49 cm$^2$ and 9 cm$^2$, respectively. If LM = 9 cm, then the length of XY is:
Triangles \(PQR\) and \(ABC\) are similar triangles. Triangle \(ABC\) have the following sides
\(AB = 12 \text{ cm}\),
\(AC = 15 \text{ cm}\),
\(CB = 21 \text{ cm}\).
Triangle \(PQR\) have the following sides:
\(PQ = 4 \text{ cm}\),
\(RQ = 7 \text{ cm}\).
What is the length of side \(PR\)?
For a pair of similar triangles, the angles made at \(Q\) is same in both the triangles \(PQR\) and \(LQJ\).
If the length of side \(QJ\) is 10 cm and \(QR\) is 5 cm, what is the ratio of their total areas?- In $\Delta ABC$ $\overline{PQ} \parallel \overline{AB}$. P and Q are on BC and CA respectively. If CQ : QA = 1 : 3 and CP = 4, then what is the value of BC?
