Problem 1 :
A rectangular gate is 3 m wide and has a 3.5 m diagonal. How high is the gate?
Solution :
(3.5)^{2} = x^{2} + 3^{2}
12.25 = x^{2} + 9
x^{2} = 12.25 - 9
x^{2} = 3.25
x = √3.25
x = 1.8
Problem 2 :
A rhombus has diagonals of length 6 cm and 8 cm. Find the length of its sides.
Solution :
In rhombus, the diagonals will intersect each other at 90 degree.
x^{2} = 3^{2} + 4^{2}
x^{2} = 9 + 16
x^{2} = 25
x = √25
x = 5
So, the side length of rhombus is 5 cm.
Problem 3 :
A man rides his bicycle due east at 16 kmh^{-1}. His son rides his bicycle due south at 20 kmh^{-1}. If they both leave point A at the same time, how far apart are they after 4 hours?
Solution :
Time taken = 4 hours
Distance covered = Time ⋅ Speed
Distance covered by bicycle in the direction east
= 4 ⋅ 16
= 64 km
Distance covered by bicycle in the direction south
= 4 ⋅ 20
= 80 km
x^{2} = 80^{2} + 64^{2}
x^{2} = 6400 + 4096
x^{2} = 10496
x = 102.44
they are 102 km apart 4 hours later.
Problem 4 :
An equilateral triangle has sides of length 6 cm. Find its area.
Solution :
6^{2} = a^{2} + 3^{2}
36 = a^{2} + 9
a = √36 - 9
a = √27
Area of triangle = (1/2) ⋅ base ⋅ height
= (1/2) ⋅ 6 ⋅ √27
= 3√27
= 15.6 cm^{2}
So, area of the equilateral triangle is 15.6 cm^{2}.
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