Q1. If triangle GHI is similar to triangle JKL, and the lengths of sides GH and JK are 5 cm and 10 cm respectively, 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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Q2. If triangle ABC is isosceles with AB = AC and angle A = 40 degrees, what is the measure of angles B and C?
Solution:
In an isosceles triangle, the base angles are equal. Therefore, angle B = angle C = (180 - 40) / 2 = 70 degrees.
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Q3. If triangle JKL is congruent to triangle MNO, which of the following is true?
Solution:
Congruent triangles have all corresponding sides and angles equal, so all statements are true.
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Q4. In triangle STU, if angle S = 30 degrees and angle T = 60 degrees, what is the measure of angle U?
Solution:
The sum of angles in a triangle is 180 degrees. Therefore, angle U = 180 - (30 + 60) = 90 degrees.
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Q5. In triangle GHI, if GH = 12 cm, HI = 16 cm, and GI = 20 cm, what is the area of triangle GHI?
Solution:
Using Heron's formula, s = (12 + 16 + 20) / 2 = 24 cm. Area = √(s(s-a)(s-b)(s-c)) = √(24(24-12)(24-16)(24-20)) = √(24*12*8*4) = 96 cm².
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Q6. In triangle STU, if angle S = 30 degrees and angle T = 70 degrees, what is the length of side SU if ST = 10 cm and TU = 15 cm?
Solution:
Using the Law of Sines: SU / sin(80) = 10 / sin(30). Therefore, SU = 10 * sin(80) / sin(30) = 10 * 0.9848 / 0.5 = 19.696 cm.
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Q7. In triangle XYZ, if XY = 7 cm, YZ = 24 cm, and XZ = 25 cm, is triangle XYZ a right triangle?
Solution:
Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2, thus triangle XYZ is a right triangle.
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Q8. In triangle MNO, if MN = 12 cm, NO = 16 cm, and MO = 20 cm, prove that triangle MNO is congruent to triangle PQR with sides PQ = 12 cm, QR = 16 cm, and PR = 20 cm.
Solution:
Both triangles have sides of equal lengths (12 cm, 16 cm, 20 cm), thus they are congruent by the SSS criterion.
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Q9. In triangle PQR, if PQ = 5 cm, QR = 12 cm, and PR = 13 cm, what type of triangle is it?
Solution:
Using the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169, which equals 13^2. Therefore, triangle PQR is a right triangle.
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Q10. If two angles of triangle ABC are 45 degrees and 55 degrees, what is the third angle?
Solution:
The sum of angles in a triangle is 180 degrees. Therefore, the third angle = 180 - (45 + 55) = 90 degrees.
