Question 5 of 11intermediate🔧 ApplyNumerical3 marks

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

Correct Answer

The other two sides are 12 cm and 5 cm.

Exercise: EXERCISE 4.2 | Q: 5 | (Page 7)
For More Understanding

Explanation

This problem involves applying the Pythagoras theorem to form a quadratic equation. The relationship between the sides (base and altitude) is given, which helps in setting up the equation. Solving the quadratic equation yields the values of the sides, and the negative root is rejected since length cannot be negative.

Solution Steps

  1. Step 1: Let the base of the right triangle be x cm. Then, the altitude = (x - 7) cm (since it is 7 cm less than the base). The hypotenuse is given as 13 cm.

  2. Step 2: Using Pythagoras theorem: (Base)² + (Altitude)² = (Hypotenuse)². Substituting the values: x² + (x - 7)² = 13².

  3. Step 3: Expanding the equation: x² + (x² - 14x + 49) = 169. Simplifying: 2x² - 14x + 49 = 169, which gives 2x² - 14x - 120 = 0.

  4. Step 4: Dividing the equation by 2: x² - 7x - 60 = 0. Solving by factorisation: (x - 12)(x + 5) = 0. So, x = 12 or x = -5.

  5. Step 5: Since length cannot be negative, x = 12. Therefore, Base = 12 cm and Altitude = 12 - 7 = 5 cm.

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