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Ex 6.4, 6 - Prove that ratio of areas of two similar triangles - Area of similar triangles

  1. Chapter 6 Class 10 Triangles
  2. Serial order wise
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Ex 6.4, 6 Exercise 6.4 Chapter 6 Class 10 CBSE NCERT Maths Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. Given: Let ∆ ABC ~∆ PQR Here AD is median Hence BD = CD = 1/2 BC Similarly, PS is median Hence QS = RS = 1/2 QR To prove: (𝑎𝑟 ∆ 𝐴𝐵𝐶)/(𝑎𝑟 ∆ 𝑃𝑄𝑅)=( 𝐴𝐷/𝑃𝑆)^2 Proof: Given ∆ ABC ~∆ PQR ∠𝐵=∠𝑄 Also, 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅 𝐴𝐵/𝑃𝑄=2𝐵𝐷/2𝑄𝑆 𝐴𝐵/𝑃𝑄=𝐵𝐷/𝑄𝑆 In ∆ 𝐴𝐵𝐷 & ∆ 𝑃𝑄𝑆 ∠𝐵=∠𝑄 𝐴𝐵/𝑃𝑄=𝐵𝐷/𝑄𝑆 ∆ 𝐴𝐵𝐷 ~ ∆ 𝑃𝑄𝑆 Hence 𝐴𝐵/𝑃𝑄=𝐴𝐷/𝑃𝑆 Now, since Δ ABC ∼ Δ PQR We know that if two triangles are similar, the ratio of their area is always equal to the square of the ratio of their corresponding side ∴ (𝑎𝑟 ∆𝐴𝐵𝐶)/(𝑎𝑟 ∆𝑃𝑄𝑅) = (𝐴𝐵/𝑃𝑄)^2 (𝑎𝑟 ∆𝐴𝐵𝐶)/(𝑎𝑟 ∆𝑃𝑄𝑅) = (𝐴𝐷/𝑃𝑆)^2 Hence proved

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