Example 1
The length of a rectangle is 4 ft. and the breadth is 3ft. What is the length of its diagonal?
Solution :
ABCD is a rectangle such that l (seg.AB) = 4 , l (seg.BC) = 3. m Ð ABC is 90^{0}.
D ABC is a right triangle.
\ l (AC)^{ 2} = l (AB)^{2} + l (BC)^{2}
= (4)^{2} + (3)^{2}
= 16 + 9
= 25
\ l (AC) = 5 ft.
The length of the diagonal is 5 ft.
Example 2
A man drives south along a straight road for 17 miles. Then turns west at right angles and drives for 24 miles where he turns north and continue driving for 10 miles before coming to a halt. What is the straight distance from his starting point to his terminal point ?
Solution :
Click here to enlarge
The man drove from A ® C ® D ® E
EB is perpendicular to AC .
\ l (AB) = 17  10 = 7 miles
l (EB) = 24 miles
D ABE is a right triangle
\ l (AE)^{2} = l (AB)^{2} + l (EB)^{2}
= (7)^{2} + (24)^{2}
= 49 + 576
= 625
\ l (AE) = 25 miles.
Example 3
D ABC is a right triangle. m Ð ACB = 90^{0} , seg.AQ bisects seg.BC at Q.
Prove that 4 l (AQ)^{2} = 4 l ( AC)^{2} + BC^{2}
Solution :
D ABC is a right triangle
\ l (AQ)^{2} = l (AC)^{2} + l (CQ)^{2}
or 4 l (AQ)^{2} = 4 l (AC)^{2} + 4 l (CQ)^{2}
= 4 l (AC)^{2} + { 2 l (CQ) }^{2}
2 l (CQ) = l (BC) as Q is the midpoint of BC
\ 4 l (AQ)^{2} = 4 l (AC)^{2} + l (BC)^{2}.
In any triangle if the square of the longest side is greater than the sum of the squares of the other two sides the triangle is an obtuse triangle. If however the square of the longest side is less than the sum of the squares of the other two sides the triangle is an acute triangle. Given the lengths of the three sides of a triangle are a, b and c where c > a and b. If c^{2} > a^{2} + b^{2} D ABC is an obtuse and if c^{2} < a^{2} + b^{2} D ABC is an acute triangle.
