Question : If the areas of two isosceles triangles with equal corresponding angles are in the ratio of $x^2:y^2$, then the ratio of their corresponding heights is:
Option 1: $x: y$
Option 2: $\sqrt{x}: \sqrt{y}$
Option 3: $x^3: y^3$
Option 4: $x^2: y^2$
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Correct Answer: $x: y$
Solution : The area of a triangle is directly proportional to the square of its height. Given, Ratio of areas = $x^2 : y^2$ $\therefore$ The ratio of their heights = $x:y$ Hence, the correct answer is $x:y$.
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Question : If the ratio of the area of two similar triangles is $\sqrt{3}:\sqrt{2}$, then what is the ratio of the corresponding sides of the two triangles?
Option 1: 9 : 4
Option 2: 3 : 2
Option 3: $\sqrt[3]{3}: \sqrt[3]{2}$
Option 4: $\sqrt[4]{3}: \sqrt[4]{2}$
Question : If the ratio of corresponding sides of two similar triangles is $\sqrt{5}: \sqrt{7},$ then what is the ratio of the area of the two triangles?
Option 1: $\sqrt[3]{5}: \sqrt{7}$
Option 2: $25: 49$
Option 3: $\sqrt{5}: \sqrt{7}$
Option 4: $5: 7$
Question : If $\cos 21^{\circ}=\frac{x}{y}$, then $(\operatorname{cosec21^{\circ}}-\cos 69^{\circ})$ is equal to:
Option 1: $\frac{x^{2}}{y\sqrt{y^{2}-x^{2}}}$
Option 2:
$\frac{y^{2}}{x\sqrt{y^{2}-x^{2}}}$
Option 3:
$\frac{y^{2}}{x\sqrt{x^{2}-y^{2}}}$
Option 4: $\frac{x^{2}}{y\sqrt{x^{2}-y^{2}}}$
Question : If $x=\frac{\sqrt{5}-\sqrt{4}}{\sqrt{5}+\sqrt{4}}$ and $y=\frac{\sqrt{5}+\sqrt{4}}{\sqrt{5}-\sqrt{4}}$ then the value of $\frac{x^2-x y+y^2}{x^2+x y+y^2}=$?
Option 1: $\frac{361}{363}$
Option 2: $\frac{341}{343}$
Option 3: $\frac{384}{387}$
Option 4: $\frac{321}{323}$
Question : Given that the ratio of the altitude of two triangles is 4 : 5, the ratio of their areas is 3 : 2, the ratio of their corresponding bases is:
Option 1: 8 : 15
Option 2: 15 : 8
Option 3: 5 : 8
Option 4: 8 : 5
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