Question : The ratio of the area of two isosceles triangles having the same vertical angle (i.e. the angle between equal sides) is 1 : 4. The ratio of their heights is:
Option 1: 1 : 4
Option 2: 2 : 5
Option 3: 1 : 2
Option 4: 3 : 4
Correct Answer: 1 : 2
Solution : $\angle$ BAC = $\angle$ QPR $\angle$ ABC = $\angle$ ACB and $\angle$ PQR = $\angle$ PRQ $\angle$ ABC = $\angle$ ACB = $\angle$ PQR = $\angle$ PRQ $\triangle$ BAC ~ $\triangle$ QPR $\frac{area(\triangle BAC)}{area(\triangle PQR)}$ = $\frac{h^2}{H^2}$ ⇒ $\frac{1}{4}$ = $\frac{h^2}{H^2}$ ⇒ $\frac{h}{H}$ = $\frac{1}{2}$ Hence, the correct answer is 1 : 2.
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Question : Two isosceles triangles have equal vertical angles and their areas are in the ratio 9 :16. Then the ratio of their corresponding heights is:
Option 1: 4.5 : 8
Option 2: 4 : 3
Option 3: 8 : 4.5
Question : The areas of the two triangles are in the ratio 4 : 3 and their heights are in the ratio 6 : 5. Find the ratio of their bases.
Option 1: 5 : 6
Option 2: 10 : 9
Option 3: 6 : 5
Option 4: 9 : 10
Question : The radii of two cylinders are in the ratio of 4 : 5 and their heights are in the ratio of 5 : 2. The ratio of their volume is:
Option 1: 9 : 4
Option 2: 2 : 1
Option 3: 9 : 7
Option 4: 8 : 5
Question : The heights of two right circular cones are in the ratio 1 : 5 and the perimeter of their bases are in the ratio 5 : 3. Find the ratio of their volumes.
Option 1: 8 : 11
Option 2: 7 : 6
Option 3: 5 : 9
Question : The ratio of the radii of two cylinders is $2:3$ and the ratio of their heights is $5:3$. The ratio of their volumes will be:
Option 1: $9:4$
Option 2: $20:27$
Option 3: $4:9$
Option 4: $27:20$
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