Key Concept: Similar Triangles Area Ratio
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Mathematically, if \(\Delta XYZ \sim \Delta LMN\), then:
\( \frac{\text{Area}(\Delta XYZ)}{\text{Area}(\Delta LMN)} = \left(\frac{XY}{LM}\right)^2 \)
Applying the Concept
Given:
- Area(\(\Delta XYZ\)) = \(49\ cm^2\)
- Area(\(\Delta LMN\)) = \(9\ cm^2\)
- LM = \(9\ cm\)
- We need to find XY.
Calculation Steps
- Set up the ratio using the areas and corresponding sides XY and LM:
\( \frac{49}{9} = \left(\frac{XY}{9}\right)^2 \)
- Take the square root of both sides to find the ratio of the sides:
\( \sqrt{\frac{49}{9}} = \frac{XY}{9} \)
\( \frac{7}{3} = \frac{XY}{9} \)
- Solve for XY by multiplying both sides by 9:
\( XY = \frac{7}{3} \times 9 \)
\( XY = 7 \times 3 \)
\( XY = 21\ cm \)
Therefore, the length of XY is 21 cm.
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?
- If in triangle \(\Delta\text{ABC}\) \(\text{AB}= 2\text{cm}\), \(\text{BC}=4\text{cm}\), and \(\text{AC}= 5\text{ cm}\) and in triangle \(\Delta\text{PQR}\) \(\text{PQ}= 12\text{cm}\), \(\text{QR}= 24\text{ cm}\), and \(\text{PR}=30\text{ cm}\), then triangles are:
- Two triangle are called similar if
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?
