The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm , find the other two sides.

Given: In right-angled triangle A B C , Hypotenuse, A C = 13 cm Let Base, B C = x cm Then, Altitude, A B = x - 7 cm Now, as per Pythagoras theorem, we have Hypotenuse 2 = Base 2 + Height 2 ⇒ A C 2 = B C 2 + A B 2 ⇒ 13 2 = x 2 + x - 7 2 ⇒ 169 = x 2 + x 2 - 14 x + 49 ⇒ 2 x 2 - 14 x - 120 = 0 ⇒ x 2 - 7 x - 60 = 0 ⇒ x 2 - 12 x + 5 x - 60 = 0 ⇒ x x - 12 + 5 x - 12 = 0 ⇒ x - 12 x + 5 = 0 ⇒ x - 12 = 0 or x + 5 = 0 ⇒ x = 12 or x = - 5 But, side of a triangle can not be negative. So, x = 12 Thus, Base of triangle, B C = 12 cm , and Altitude of triangle, A B = 12 - 7 = 5 cm Hence, the required two sides of the triangle are 5 cm and 12 cm .

The sum of a number and its reciprocal is 26 5 . Find the number.

Let the number be x so the reciprocal of the number is 1 x . According to the question, x + 1 x = 26 5 ⇒ x 2 + 1 x = 26 5 ⇒ 5 x 2 + 1 = 26 x ⇒ 5 x 2 + 5 - 26 x = 0 ⇒ 5 x 2 - 26 x + 5 = 0 By using quadratic formula, we get x = - - 26 ± - 26 2 - 4 × 5 × 5 2 × 5 ⇒ x = 26 ± 676 - 100 10 ⇒ x = 26 ± 576 10 ⇒ x = 26 ± 24 10 So, ⇒ x = 26 + 24 10 = 5 Or, x = 26 - 24 10 = 1 5 So, the number is 5 or 1 5 .

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An express train takes $1$ hour less than a passenger train to travel $132 km$ between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is $11 km / h$ more than that of the passenger train, find the average speed of the two trains.

Important Questions on Quadratic Equations

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Mathematics>Algebra>Quadratic Equations>Word Problems on Quadratic Equation

The altitude of a right triangle is

= 44 km / hour .

less than its base. If the hypotenuse is

13 cm

, find the other two sides.

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Mathematics>Algebra>Quadratic Equations>Word Problems on Quadratic Equation

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20 years$ . Four years ago, the product of their ages in years was $48 .$

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Mathematics>Algebra>Quadratic Equations>Word Problems on Quadratic Equation

A dealer sells an article for

₹ 16

and loses as much per cent as the cost price of the article. Find the cost price of the article.

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Mathematics>Algebra>Quadratic Equations>Word Problems on Quadratic Equation

A motor boat whose speed is $18 km / hour$ in still water takes $1 hour$ more to go $24 km$ upstream than to return downstream to the same spot. If the speed of the stream is $x km / hour$ , find $x$ .