(a) An isosceles right-angled triangle has area 200 cm². What is the length of its hypotenuse?
(b) The perimeter of a triangle is 540 m and its sides are in the ratio 12 : 25 : 17. Find the area of the triangle.
(c) Find the area of triangle whose sides are 5 cm, 12 cm and 13 cm. Also, find the length of its altitude corresponding to the longest side.

Question

(a) An isosceles right-angled triangle has area 200 cm². What is the length of its hypotenuse?
(b) The perimeter of a triangle is 540 m and its sides are in the ratio 12 : 25 : 17. Find the area of the triangle.
(c) Find the area of triangle whose sides are 5 cm, 12 cm and 13 cm. Also, find the length of its altitude corresponding to the longest side.

KnowledgeBoat · Accepted Answer

(a) In an isosceles right-angled triangle, the two perpendicular sides (base and height) are of equal length. The area is given by: Area = Let the base of the triangle be x. Since it is an isosceles right-angled triangle, the height is also x. Area = ⇒ 200 = ⇒ 200 2 = ⇒ = 400 ⇒ ⇒ = 20 Now, using the Pythagoras Theorem, the hypotenuse h is given by: h2 = base2 + height2 ⇒ h2 = (20)2 + (20)2 ⇒ h2 = 400 + 400 ⇒ h2 = 800 ⇒ h = ⇒ h = 20 Hence, the length of the hypotenuse is 20 cm. (b) The given ratio of sides is 12 : 25 : 17. Let the sides of the triangle be 12x, 25x and 17x. The perimeter of the triangle = Sum of all sides ⇒ 540 m = 12x + 25x + 17x ⇒ 540 m = 54x ⇒ x = m ⇒ x = 10 m So, the actual lengths of the sides are: 12x = 12 10 = 120 m 25x = 25 10 = 250 m 17x = 17 10 = 170m Using Heron's formula, semi-perimeter (s), The area of the triangle, Hence, the area of the triangle is 9,000 m2. (c) Using Heron's formula, The area of the triangle, Area of the triangle = x base x altitude Let the altitude corresponding to the longest side (13 cm) be h. ⇒ x 13 x h = 30 ⇒ 13 x h = 30 x 2 ⇒ 13 x h = 60 ⇒ h = ⇒ h = 4.61 cm Hence, area of the triangle is 30 cm2 and the altitude is 4.61 cm.

Mathematics

Mensuration

Related Questions

Find the median of 17, 26, 60, 45, 33, 32, 29, 34 and 56. If 26 is replaced by 62, what will be the new median ?

The following data have been arranged in ascending order of magnitude.
63, 66, 69, x, x + 2, 76, 89 and 103.
If the median of the given data is 71, find the value of x.

(a) The diagonals of a rhombus are 24 cm and 10 cm. Calculate its area and perimeter.
(b) The diagonals of a field in the form of a quadrilateral are 106 m and 80 m and intersect each other at right angles. Find the cost of cultivating the field at the rate of ₹ 25.50 per 100 m².

If the difference between the two sides of a right-angled triangle is 2 cm and the area of the triangle is 24 cm²; find the perimeter of the triangle.