Q1. If triangle GHI is similar to triangle JKL and the length of side GH is 5 cm while side JK is 10 cm, what is the ratio of the areas of the two triangles?
Solution:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (10/5)^2 = 2^2 = 4, so the ratio of the areas is 1:4.
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Q2. In triangle ABC, if AB = 12 cm, AC = 16 cm, and angle A = 60 degrees, what is the length of BC?
Solution:
Using the Law of Cosines: BC² = AB² + AC² - 2(AB)(AC)cos(A) = 12² + 16² - 2(12)(16)(0.5) = 144 + 256 - 192 = 208. Therefore, BC = √208 = 14.42 cm.
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Q3. If triangle XYZ is similar to triangle PQR and the length of XY is 5 cm and PQ is 10 cm, what is the ratio of their areas?
Solution:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (5/10)² = 1/4.
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Q4. In triangle GHI, if angle G = 45 degrees and angle H = 45 degrees, what type of triangle is it?
Solution:
A triangle with two equal angles is an isosceles triangle.
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Q5. In triangle STU, if ST = 12 cm, TU = 16 cm, and SU = 20 cm, what is the perimeter of triangle STU?
Solution:
The perimeter of triangle STU is ST + TU + SU = 12 + 16 + 20 = 48 cm.
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Q6. In triangle ABC, if the lengths of sides AB and AC are equal, what type of triangle is ABC?
Solution:
A triangle with two equal sides is classified as an isosceles triangle.
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Q7. In triangle DEF, if DE = 6 cm, DF = 8 cm, and EF = 10 cm, what is the area of the triangle?
Solution:
Using Heron's formula, s = (6 + 8 + 10) / 2 = 12. Area = √[s(s-a)(s-b)(s-c)] = √[12(12-6)(12-8)(12-10)] = √[12*6*4*2] = √576 = 24 cm².
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Q8. If the lengths of the sides of triangle PQR are in the ratio 3:4:5, what type of triangle is it?
Solution:
A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.
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Q9. If triangle VWX is similar to triangle YZ, and the length of side VW is 9 cm while side YZ is 3 cm, what is the scale factor from triangle YZ to triangle VWX?
Solution:
The scale factor from triangle YZ to triangle VWX is 9/3 = 3, so the scale factor is 3:1.
