The relationship between a perpendicular, hypotenuse, and base of a triangle is stated by the Pythagoras Theorem. This can also be called as the right-angled triangle theorem. According to the Pythagoras theorem, if the sum of the square of perpendicular and the square of the base is equal to the square of the hypotenuse, then it is known as a right-angled triangle. It can be expressed as follows:
If Hypotenuse2 = Perpendicular2 + Base2, then angle = 90˚
The side that is opposite to the angle is known as the hypotenuse, the adjacent side to the right angle, and is connected to the hypotenuse is the perpendicular and base is the adjacent side of the right angle. The base is also called as the legs of the hypotenuse. The length of any side of the triangle can be found if we have two known values of the triangle. In this triangle, one angle is equal to 90˚ and the other two angles are equal to the sum of 90˚. You can always change the base and the perpendicular sides depending on how we consider the acute angle. The two sides of the triangle, i.e., base, hypotenuse, and the perpendicular are considered to be Pythagoras triplets and it is known as Pythagorean triangle if all the sides are integers.
Theorem: In a right-angled triangle, if the sum of the square of two sides is equal to the square of one side, then the right angle is the angle that is opposite to the first side.
Proof of right angle triangle.
To prove ∠Q = 90˚ .
Proof: We have a triangle in which PR2 = PQ2 + QR2
We need to prove that ∠Q = 90˚ .
To prove the above condition, we need to first construct a triangle ΔLMN such that M is the right angle and is equal to 90˚ .
LM² = LN²+ NM² (According to Pythagoras theorem,as ∠N = 90°)
or, LM² =PQ² + QR² ( By construction ) …… (1)
We know that;
AC² = PQ² + QR² ( Which is given ) …………(2)
So, AC = LM [ From equation (1) and (2) ]
Now, in Δ ABC and Δ LMR,
PQ = LM ( By construction )
QR = MN ( By construction )
PR = LN [ Proved above ]
So, Δ PQR ≅ Δ LMN ( By SSS congruence )
Therefore, ∠Q = ∠M ( CPCT )
But, ∠M = 90° ( By construction )
So, ∠Q = 90°
Hence the theorem is proved.
The formula of the right-angled triangle goes by:
( Hypotenuse ) 2 = ( Opposite ) 2 + ( Adjacent )2
If p, q and r are hypotenuse, adjacent, and the opposite of a right-angles triangle, then:
p2 = r2 + q2
or
p = \[\sqrt{r^{2} + q^{2}}\]
This means, the root of the sum of the squares of the base and the perpendicular.
Half of the product of the base and the perpendicular is equal to the area of the triangle.
Right Angle Triangle Formula is:
A = \[\frac{1}{2}\] x base x height
Here,
B = base
H = height of the triangle
Question 1: The angle ∠JLK = 90˚ and LP is perpendicular to K. Prove that KJ2 / JL2 = KP / JP.
Solution: We can see that;
△ JJL ~ △ JLK
According to the property of similar triangles we have:
JL / JK = JP / JL
or it can also be written as, JL2 = JK . JP ……….(1)
Similarly, we have △ KPL ~ KLJ
So we have, KP / KL = KL / KJ
Or it can also be written as;
KL2 = KJ . KP ……….. (2)
By dividing the equations (2) by (1) we can deduce that:
LL2 / JL2 = ( LJ . LP ) / ( JL . JP ) = LP / JP
Hence, it proved.
Let’s study some right-angle triangle examples.
Question 2: If the length of the hypotenuse and the base is 6 cm and 5 cm of the right triangle. Find the height of the triangle.
Solution: We know the formula of the right-angle triangle. It goes by:
( Hypotenuse ) 2 = ( Opposite ) 2 + ( Adjacent )2
( 6 )2 = ( 5 )2 + ( Perpendicular )2
36 = 25 + ( P )2
36 - 25 = ( P )2
P = \[\sqrt{9}\] = 3 cm
Hence, the height of the perpendicular is 3 cm.
Question 3: If the length of the hypotenuse and the base is 10 cm and 6 cm of the right triangle. Find the height of the triangle.
Solution: We know the formula of the right-angle triangle. It goes by:
( Hypothenuse ) 2 = ( Opposite ) 2 + ( Adjacent )2
( 10 )2 = ( 6 )2 + ( Perpendicular )2
100 = 36 + ( P )2
100 - 36 = ( P )2
P = \[\sqrt{64}\] = 8 cm
Hence, the height of perpendicular is 8 cm.
1) How to Prove a Right Angle?
The relationship between a perpendicular, hypotenuse, and base of a triangle is stated by the Pythagoras Theorem. This can also be called as the right-angled triangle theorem. According to the Pythagoras theorem, if the sum of the square of perpendicular and the square of the base is equal to the square of the hypotenuse, then it is known as a right-angled triangle. It can be expressed as follows:
(Hypotenuse)^{2} = (Perpendicular)^{2} + (Base)^{2}, then angle = 90˚
Theorem: In a right-angled triangle, if the sum of the square of two sides is equal to the square of one side, then the right angle is the angle that is opposite to the first side.
To prove ∠Q = 90˚.
Proof: We have a triangle in which PR^{2} = PQ^{2} + QR^{2}
We need to prove that ∠Q = 90˚.
To prove the above condition, we need to first construct a triangle ΔLMN such that M is the right angle and is equal to 90˚.
LM^{2} = LN^{2}+ NM^{2} (According to Pythagoras theorem,as ∠N = 90°)
or, LM^{2} = PQ^{2} + QR^{2} ( By construction ) …… (1)
We know that;
AC^{2} = PQ^{2} + QR^{2} ( Which is given ) …………(2)
So, AC = LM [ From equation (1) and (2) ]
Now, in Δ ABC and Δ LMR,
PQ = LM ( By construction )
QR = MN ( By construction )
PR = LN [ Proved above ]
So, Δ PQR ≅ Δ LMN ( By SSS congruence )
Therefore, ∠Q = ∠M ( CPCT )
But, ∠M = 90° ( By construction )
So, ∠Q = 90°
Hence the theorem is proved.
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