The perimeter of an isosceles triangle is 44 cm. The ratio of the equal side to its base is 4 : 3. Find the length of the longest altitude
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Answers
Given that : Perimeter of an isosceles triangle is 44 cm.
Ratio of equal sides to it's base is 4:3
Need to find : Length of longest altitude.
Concept :
- Perimeter of a triangle is sum of all sides
- Altitude lying on the smallest side is the longest altitude.
- According to Pythagoras theorem :
Hypotenuse² = Altitude² + Base²
Solution :
⇒ Let the equal sides be 4x, 4x and base be 3x according to the given ratios. We can form an equation now as we're given the perimeter and we've supposed the sides.
~We can write it as -
→ 4x + 4x + 3x = 44 cm
→ 11x = 44 cm
→ x = 44/11
→ x = 4
Henceforth, the sides are
Equal sides = 4×4 = 16 cm
Base = 4×3 = 12 cm
⇒ The smallest side is 12 cm which is the base, as you can see in the diagram attached. We can apply the Pythagoras theorem here.
→ Hypotenuse² = Altitude² + Base²
→ 16² = Altitude² + 6²
→ 256 = Altitude² + 36
→ Altitude² = 256 - 36
→ Altitude = √220
→ Altitude = 14.83 ( approximately )
- Henceforth, the length of longest altitude is 14.83 approximately.
Given : The perimeter of an isosceles triangle is 44cm. The ratio of the equal side to it's base is 4:3 .
To Find : the length of the longest altitude
Solution:
Let say equal sides are 4x cm
Then base = 3x cm
Perimeter = 4x + 4x + 3x = 44
=> 11x = 44
=> x = 4
Hence equal sides = 4 * 4 = 16 cm
Base = 3 * 4 = 12 cm
As base is shorter than equal sides hence longest altitude will be at base
Altitude at base bisects it
Hence half of base = 12/2 = 6 cm
Length of altitude = √16² - 6²
= √220
= 2√55 cm
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