Similarity — Maharashtra (SSC) Class 10 Mathematics Solutions (Free)
Free step-by-step Maharashtra (SSC) Class 10 Mathematics solutions for "Similarity" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Maharashtra (SSC) Class 10 Mathematics solutions for "Similarity" — important questions with detailed answers, download PDF for boar…
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Q1: Triangles ABC and DEF are similar. If AB = 6 cm, BC = 8 cm, and DE = 9 cm, find EF.
Step 1: Understand similarity. If triangles ABC and DEF are similar, corresponding sides are proportional.
Step 2: Identify corresponding sides. AB corresponds to DE, BC corresponds to EF, AC corresponds to DF.
Step 3: Set up proportion. AB/DE = BC/EF.
Step 4: Substitute known values. 6/9 = 8/EF.
Step 5: Cross multiply. 6 * EF = 8 * 9.
Step 6: Solve for EF. 6 * EF = 72, so EF = 72/6 = 12 cm.
Step 7: Verify ratio. AB/DE = 6/9 = 2/3, BC/EF = 8/12 = 2/3. Ratios equal, confirming similarity.
Answer:…
Q2: In triangle ABC, DE is parallel to BC. If AD = 4 cm, DB = 6 cm, and EC = 9 cm, find AE.
Step 1: Apply Basic Proportionality Theorem (BPT). If DE || BC, then AD/DB = AE/EC.
Step 2: Substitute known values. 4/6 = AE/9.
Step 3: Cross multiply. 4 * 9 = 6 * AE.
Step 4: Solve for AE. 36 = 6 * AE, so AE = 36/6 = 6 cm.
Step 5: Verify. AD/DB = 4/6 = 2/3, AE/EC = 6/9 = 2/3. Ratios equal, confirming DE || BC.
Answer: AE = 6 cm.
Q3: Two similar triangles have areas 25 cm^2 and 100 cm^2. If a side of smaller triangle is 5 cm, find corresponding side of larger triangle.
Step 1: Recall area relationship for similar triangles. Ratio of areas = (Ratio of corresponding sides)^2.
Step 2: Set up ratio. Area1/Area2 = (Side1/Side2)^2.
Step 3: Substitute. 25/100 = (5/Side2)^2.
Step 4: Simplify. 1/4 = (5/Side2)^2.
Step 5: Take square root. 1/2 = 5/Side2.
Step 6: Solve for Side2. Side2 = 5 * 2 = 10 cm.
Step 7: Verify. (5/10)^2 = (1/2)^2 = 1/4. Area ratio = 25/100 = 1/4. Correct.
Answer: Corresponding side of larger triangle is 10 cm.
Q4: State and prove the Basic Proportionality Theorem (Thales Theorem).
Statement: If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides in the same ratio.
Proof: Consider triangle ABC with DE parallel to BC, where D is on AB and E is on AC.
Step 1: To prove: AD/DB = AE/EC.
Step 2: Construct CF parallel to DE, meeting extended BA at F.
Step 3: Since DE || BC and CF || DE, then CF || BC.
Step 4: Since BCED is a parallelogram (opposite sides parallel), DE = BC and BD = EC.
Step 5: Consider triangles ADE and…
Q5: Triangles PQR and XYZ are similar with ratio of areas 16:25. If PQ = 12 cm, find XY.
Step 1: Use area ratio for similar triangles. Area(PQR)/Area(XYZ) = (PQ/XY)^2.
Step 2: Substitute known values. 16/25 = (12/XY)^2.
Step 3: Take square root of both sides. sqrt(16/25) = 12/XY.
Step 4: Simplify. 4/5 = 12/XY.
Step 5: Cross multiply. 4 * XY = 5 * 12.
Step 6: Solve. 4 * XY = 60, so XY = 60/4 = 15 cm.
Step 7: Verify. (12/15)^2 = (4/5)^2 = 16/25. Area ratio confirmed.
Answer: XY = 15 cm.
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