Problem Analysis:
- We are given two similar triangles, $\Delta$XYZ and $\Delta$LMN.
- Their areas are Area($\Delta$XYZ) = 49 cm$^2$ and Area($\Delta$LMN) = 9 cm$^2$.
- A side length LM = 9 cm is given for $\Delta$LMN.
- We need to find the length of the corresponding side XY in $\Delta$XYZ.
Similar Triangles Area-Side Relationship
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Mathematically, this is expressed as:
$ \frac{\text{Area}(\Delta \text{XYZ})}{\text{Area}(\Delta \text{LMN})} = \left(\frac{\text{XY}}{\text{LM}}\right)^2 $Calculating Side XY
- Substitute known values into the formula: $ \frac{49 \text{ cm}^2}{9 \text{ cm}^2} = \left(\frac{\text{XY}}{9 \text{ cm}}\right)^2 $
- Take the square root of both sides: $ \sqrt{\frac{49}{9}} = \frac{\text{XY}}{9 \text{ cm}} $ $ \frac{7}{3} = \frac{\text{XY}}{9 \text{ cm}} $
- Solve for XY: Multiply both sides by 9 cm: $ \text{XY} = \frac{7}{3} \times 9 \text{ cm} $ $ \text{XY} = 7 \times 3 \text{ cm} $ $ \text{XY} = 21 \text{ cm} $
Final Answer Determination
The calculated length of side XY is 21 cm.
Comparing this with the options:
- Option A: 21 cm
- Option B: 7 cm
- Option C: 49 cm
- Option D: 14 cm
The calculated length matches Option A.
Similar Questions
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
- 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.
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?
- The altitudes of two similar triangles are 4 cm and 6 cm. If the area of the larger triangle is $36\text{ cm}^2$, what will be the area of the other one?
