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If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel
to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.?
to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.?
Most Upvoted Answer
If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively ...
Understanding the Problem
In triangle ∆PQR, we have points S and T on sides PQ and PR respectively. The given lengths are:
- PT = 2 cm
- TR = 4 cm
- ST is parallel to QR.
Using the Properties of Similar Triangles
Since ST is parallel to QR, it follows that triangles ∆PST and ∆PQR are similar by the Basic Proportionality Theorem (also known as Thales' theorem).
Finding the Length of PR
To find the total length of PR:
- PR = PT + TR = 2 cm + 4 cm = 6 cm.
Calculating the Ratio of the Sides
Using the lengths we have:
- The ratio of the sides PT to PR is:
\[
\frac{PT}{PR} = \frac{2}{6} = \frac{1}{3}
\]
Finding the Ratio of the Areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:
\[
\text{Area Ratio} = \left(\frac{PT}{PR}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}
\]
Conclusion
The ratio of the areas of ∆PST to ∆PQR is:
- **Area Ratio = 1 : 9**
This means that the area of triangle ∆PST is one-ninth the area of triangle ∆PQR.
In triangle ∆PQR, we have points S and T on sides PQ and PR respectively. The given lengths are:
- PT = 2 cm
- TR = 4 cm
- ST is parallel to QR.
Using the Properties of Similar Triangles
Since ST is parallel to QR, it follows that triangles ∆PST and ∆PQR are similar by the Basic Proportionality Theorem (also known as Thales' theorem).
Finding the Length of PR
To find the total length of PR:
- PR = PT + TR = 2 cm + 4 cm = 6 cm.
Calculating the Ratio of the Sides
Using the lengths we have:
- The ratio of the sides PT to PR is:
\[
\frac{PT}{PR} = \frac{2}{6} = \frac{1}{3}
\]
Finding the Ratio of the Areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:
\[
\text{Area Ratio} = \left(\frac{PT}{PR}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}
\]
Conclusion
The ratio of the areas of ∆PST to ∆PQR is:
- **Area Ratio = 1 : 9**
This means that the area of triangle ∆PST is one-ninth the area of triangle ∆PQR.
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If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.?
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If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.?.
If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Fig. 𝑆 and 𝑇 are points on the sides 𝑃𝑄 and 𝑃𝑅, respectively of ∆ 𝑃𝑄𝑅, such that 𝑃𝑇 = 2 𝑐𝑚, 𝑇𝑅 = 4 and 𝑆𝑇 is parallel to 𝑄𝑅. Find the ratio of the areas of ∆𝑃𝑆𝑇 and ∆𝑃𝑄𝑅.?.
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