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Quadratic Equations

Long Answer Type

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

Let the base (BC) of the right triangle = x cm
So, altitude (AB) = (x – 7) cm
And hypotenuse (AC) = 13 cm

According to Pythagoras theorem,
AC² = AB² + BC²⇒ (13)² = (x – 7)² + (x)²⇒ 169 = x² + 49 – 14x + x²⇒ 169 = 2x² –14x + 49
⇒ 2x² – 14x – 120 = 0
⇒ x² – 7x – 60 = 0
⇒ x² – 12x + 5x – 60 = 0
⇒ x(x – 12) + 5(x – 12) = 0
⇒ (x + 5) (x – 12) = 0
⇒ x + 5 = 0
or x – 12 = 0
⇒ x = – 5
or x = 12
Since side of triangle is never negative
So, x = 12
Now required sides of the triangle are
AB = x – 7
= 12 – 7 = 5 cm and BC = x = 12 cm.

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192. The sum of the areas two squares is 640 m². If the difference in their perimeter be 64 m, Find the sides of the two squares.

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193. If twice the area of a smaller square is subtracted from the area of a larger square, the result is 14 sq cm. However, if twice the area of larger square is added to three times the area of the smaller square, the result is 203 sq. cm. Find the sides of the two squares.

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Short Answer Type

194. The hypotenuse of a right triangle is

3 square root of 5 space cm.

If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15 cm. Find the length of each side.

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Long Answer Type

195. A farmer wishes to grow a 100 m²rectangular vegetable garden. Since he has with the only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the four the side-fence. Find the dimension of his garden.

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196. There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at Rs. 2.75 and Rs. 1.50 per square metre, respectively, is Rs. 4904. Find the width of the gravel path.

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197. Two pipes running together can fill a cistern in

space space 3 1 over 13 space minutes.

If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.

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198. A person on tour has Rs. 360 for his expenses. If he extends his tour for 4 days, he has to cut down his daily expenses by Rs. 3. Find the original duration of the tour.

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199. Rs. 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs. 30 less. Find the original number of persons.

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200. A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more books for the same amount, each book would have cost Rs. 1 less. How many books did he buy?

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