discrete math

## If the right angled triangle t, with sides of length a and b and hypotenuse of length c, has area equal to c^2/4, then t is an isosceles triangle.

# 3 Answers

Since this is for Discrete Math, I'm assuming that what you are trying to do is prove this statement by the laws of logic.

Let r be "The right angled triangle t has side lengths a and b, hypotunese c, with area c^{2}/4." This is our
**sufficient **part of the conditional.

Let i be "The right angled triangle t is an isosceles triangle." This is our **
necessary **part of the conditional.

Putting your statement into logic format, we have the conditional: (r --> i). Which means if r, then i.

There are 3 different ways to prove conditionals: by direct proof, contrapositive, and contradiction. Direct proof is when you assume that the sufficient side is true, then prove that the necessary side is also true. In this case we would be assuming r is
true, and arriving at the conclusion that i is also true. Contrapositive is when you assume the necessary part is false, and show that when the necessary side is false then the sufficient side is also false. In this case we would be assuming (~i --> ~r), if
~i (i is false), then ~r (r is false). Contrapositive works as a proof for the conditional since (r --> i) is equivalent to (~i --> ~r) by the laws of logic. Contradiction is when you assume that the sufficient side is true *and* the necessary side
is false, so in this case we'd assume r and ~i (r.~i), and go on with our proof to arrive at a contradiction or an absurdity in the proof. Such as r = ~r.

One mistake that many people make in proofs is by assuming that the necessary part is true, then they go on to prove that when the necessary side is true, then the sufficient side is true. This is a big no-no in logic. In this case, by already assuming the
triangle is isosceles and starting our proof from that, we would be proving (i-->r); this is
**not **equivalent to (r-->i).

Let's prove this statement by direct proof, since I believe that is the simplest way.

What do we know about i? We know that a=b if and only if it is an isosceles triangle. This means that if we can show that if r, then a=b, then we have shown that if r, then i. In the rules of logic this is called the Chain Rule.

**Remember: We cannot assume that a=b, because this is what we are trying to prove.**

So let's try to show that if r, then a=b!

So we start by assuming r is true. What do we know about r? It is a right triangle, with sides a and b, and hypotenuse c. So we know that the Pythagorean theorem holds and a^{2}+b^{2}=c^{2}. What else do we know about r? It's area
is c^{2}/4. The area of a right triangle's formula is A=(1/2)bh, where b is the base and h is the height. In this case we can let b = the base and a = the height. Plugging in our information we get the formula c^{2}/4 = (1/2)ab. Let's replace
c^{2} with a^{2}+b^{2} in our area formula. So we get:

(a^{2}+b^{2})/4=(1/2)ab

Multiply both sides by 4 to get:

a^{2}+b^{2}=2ab

Solving for 0 we get:

a^{2}-2ab+b^{2}=0

And factoring the polynomial we get:

(a-b)^{2}=0

Take the square root of both sides:

±(a-b)=0

a=b

So we have proven that a=b. Since we know that when a=b, we have an isosceles triangle, we have now shown by direct proof that if the right triangle t with sides a and b, hypotenuse c, and area c^{2}/4, then the right triangle t is an isosceles triangle.

Given: Angle t is right angle, a = b ,

Since t is Rt angle, side a is perpendicular to side b or equivalent to height

Therefore, area (A) = 1/2 a. b = 1/2 a^2 or 1/2 b^2 ... .... .... ..... ..... (1)

Also, c^2= a^2+ b^2 = 2a^2 or 2b^2 is given for Rt angled triangle.

Therefore, by dividing both sides by 2 we get a^2 = 1/2 c^2. ... .... ... (2)

By substitution this value from (2) in (1) above, we get

A = 1/2 . 1/2 c^2 = 1/4 c^2

i hope it is clear enough.

Bhadra

- Bhadra P. 10 minutes ago

I'm not sure what the actual question is, but if your goal is to understand why this is true, then I think I can help.

First note that in all right triangles the following are true:

a^{2 }+ b^{2 }= c^{2}

Area = (1/2)(a)(b)

Next observe:

a=b if and only if t is isosceles.

This allows the substitution of a for b in the original equations.

a^{2} + a^{2} = c^{2}

Area = (1/2) a^{2}

Solving the top equation for a^{2 }yields

a^{2 }= (1/2) c^{2}

which when substituted into the area formula gives

Area = (1/2)(1/2) c^{2 }= (1/4)c^{2}

I hope that was helpful!

## Comments

Given: Angle t is right angle, a = b ,

Since t is Rt angle, side a is perpendicular to side b or equivalent to height

Therefore, area (A) = 1/2 a. b = 1/2 a^2 or 1/2 b^2 ... .... .... ..... ..... (1)

Also, c^2= a^2+ b^2 = 2a^2 or 2b^2 is given for Rt angled triangle.

Therefore, by dividing both sides by 2 we get a^2 = 1/2 c^2. ... .... ... (2)

By substitution this value from (2) in (1) above, we get

A = 1/2 . 1/2 c^2 = 1/4 c^2

i hope it is clear enough.

Bhadra Panta

There is the additional necessary condition that the triangle is isosceles for this explanation to be valid. The fact that there is a right triangle does not guarantee that a=b. (Note 3,4,5 right triangles for example.) The right angle simply validates the Pythagorean theorem for the area of the triangle.

Comment