A right triangle with sides a, b, c forming arithmetic sequence. The perimeter of that triangle is p cm and its area is q cm squared. Given that 3p = 2q, what is the value of b?

Value of B is Buhari.... chaii upon all the money my people spent for me to learn maths still e no gree enter.

How this math wan take put food for ya table this morning?

If you can't help, just say something reasonable or look and pass.

Chnbanc:
How this math wan take put food for ya table this morning?

b= 10 (Answer)

a=9, c=11

Since the sides form an arithmetic sequence, then c will be the longest (hypotenus) while b and a will be the base and height.

Since we are looking for b, the sequence will be: b-1, b, b+1
Perimeter, p= (b-1) + (b) + (b+1) = 3b
Area= 1/2 (bh)= 1/2 (b)(b-1)
Area,q= (b^2 - b)/2

Since 3p = 2q
3(3b)= 2(b^2-b)/2
9b=b^2-b
b^2-10b= 0
b=0 or b=10

For an obvious reason, we have to consider 10 and leave out 0.

Therefore, b=10

C’est fini.

zealousayo:
b= 10 (Answer)

a=9, c=11

Since the sides form an arithmetic sequence, then c will be the longest (hypotenus) while b and a will be the base and height.

Since we are looking for b, the sequence will be: b-1, b, b+1
Perimeter, p= (b-1) + (b) + (b+1) = 3b
Area= 1/2 (bh)= 1/2 (b)(b-1)
Area,q= (b^2 - b)/2

Since 3p = 2q
3(3b)= 2(b^2-b)/2
9b=b^2-b
b^2-10b= 0
b=0 or b=10

For an obvious reason, we have to consider 10 and leave out 0.

Therefore, b=10

C’est fini.

Thank you so much. I need to stay in touch with you. Can you give me your WhatsApp contact?

Timijo:
Thank you so much. I need to stay in touch with you. Can you give me your WhatsApp contact?

One more questions

We have 10b = b^2
How did we arrive at b =10?

zealousayo:
b= 10 (Answer)

a=9, c=11

Since the sides form an arithmetic sequence, then c will be the longest (hypotenus) while b and a will be the base and height.

Since we are looking for b, the sequence will be: b-1, b, b+1
Perimeter, p= (b-1) + (b) + (b+1) = 3b
Area= 1/2 (bh)= 1/2 (b)(b-1)
Area,q= (b^2 - b)/2

Since 3p = 2q
3(3b)= 2(b^2-b)/2
9b=b^2-b
b^2-10b= 0
b=0 or b=10

For an obvious reason, we have to consider 10 and leave out 0.

Therefore, b=10

C’est fini.

One more questions

We have 10b = b^2
How did we arrive at b =10?

Timijo:
Thank you so much. I need to stay in touch with you. Can you give me your WhatsApp contact?

All right, no problem... Can you send me a private mail?

Timijo:
One more questions

We have 10b = b^2
How did we arrive at b =10?

Even if you divide both sides by b, the answer is 10 (without resorting to a quadratic equation).

Timijo:
One more questions

We have 10b = b^2
How did we arrive at b =10?

10b=b²
10(b)=b(b)
Divide both side by b and you have
b=10.

zealousayo:

All right, no problem... Can you send me a private mail?

I sent you an email.

Timijo:
I sent you an email.

Okay, I've responded to it.

zealousayo:
b= 10 (Answer)

a=9, c=11

Since the sides form an arithmetic sequence, then c will be the longest (hypotenus) while b and a will be the base and height.

Since we are looking for b, the sequence will be: b-1, b, b+1
Perimeter, p= (b-1) + (b) + (b+1) = 3b
Area= 1/2 (bh)= 1/2 (b)(b-1)
Area,q= (b^2 - b)/2

Since 3p = 2q
3(3b)= 2(b^2-b)/2
9b=b^2-b
b^2-10b= 0
b=0 or b=10

For an obvious reason, we have to consider 10 and leave out 0.

Therefore, b=10

C’est fini.

Approach is somewhat correct but this is wrong. OP if you have not submitted this then don't.

You made a critical assumption which is that the AP is of difference 1. That was never stated anywhere in the question.

If you would like to verify your answer then the three side lengths should adhere to Pythagoras since we are dealing with a right-angled triangle. (9²)+(10²)≠(11²).

Here is the correct solution.

The three sides a,b,c can be written as a-d, a, a+d respectively since it's in an A.P

p=(a-d)+a+(a+d)
p=3a

The area is the product the two smaller sides divided by two(this is from 1/2*base*height)

q=(1/2)*(a-d)*a
q=(a²-ad)/2

Note: Since we are dealing with a right angled triangle:

(a-d)² + a² = (a+d)²

Basic simplification gives you
a=4d...(1)

We'll need this later.

Now,
3p=2q
9a=a²-ad

Substituting d with a/4 from Equation (1)
9a=a²-(a*a/4)
9a=a²-(a²/4)

=> (3/4)a² = 9a
=>(3/4)a=9
=>a=12

Therefore the three sides of the triangle are:
a-d=9

a=12

a+d=15

We can check that these are truly Pythagorean triples since

9²+12²=15²

The question asked for only b though so

b=12

Najdorf:

Approach is somewhat correct but this is wrong. OP if you have not submitted this then don't.

You made a critical assumption which is that the AP is of difference 1. That was never stated anywhere in the question.

If you would like to verify your answer then the three side lengths should adhere to Pythagoras since we are dealing with a right-angled triangle. (9²)+(10²)≠(11²).

Here is the correct solution.

The three sides a,b,c can be written as a-d, a, a+d respectively since it's in an A.P

p=(a-d)+a+(a+d)
p=3a

The area is the product the two smaller sides divided by two(this is from 1/2*base*height)

q=(1/2)*(a-d)*a
q=(a²-ad)/2

Note: Since we are dealing with a right angled triangle:

(a-d)² + a² = (a+d)²

Basic simplification gives you
a=4d...(1)

We'll need this later.

Now,
3p=2q
9a=a²-ad

Substituting d with a/4 from Equation (1)
9a=a²-(a*a/4)
9a=a²-(a²/4)

=> (3/4)a² = 9a
=>(3/4)a=9
=>a=12

Therefore the three sides of the triangle are:
a-d=9

a=12

a+d=15

We can check that these are truly Pythagorean triples since

9²+12²=15²

The question asked for only b though so

b=12

You're right, I shouldn't have made that assumption.
@Timijo... Follow this approach.

zealousayo:

You're right, I shouldn't have made that assumption.
@Timijo... Follow this approach.

Okay, thanks.

Thank you for the correction. Please, I will need to keep in touch with you. I need help with some questions. Please send me your WhatsApp number via email. I have sent you an email.

Najdorf:

Approach is somewhat correct but this is wrong. OP if you have not submitted this then don't.

You made a critical assumption which is that the AP is of difference 1. That was never stated anywhere in the question.

If you would like to verify your answer then the three side lengths should adhere to Pythagoras since we are dealing with a right-angled triangle. (9²)+(10²)≠(11²).

Here is the correct solution.

The three sides a,b,c can be written as a-d, a, a+d respectively since it's in an A.P

p=(a-d)+a+(a+d)
p=3a

The area is the product the two smaller sides divided by two(this is from 1/2*base*height)

q=(1/2)*(a-d)*a
q=(a²-ad)/2

Note: Since we are dealing with a right angled triangle:

(a-d)² + a² = (a+d)²

Basic simplification gives you
a=4d...(1)

We'll need this later.

Now,
3p=2q
9a=a²-ad

Substituting d with a/4 from Equation (1)
9a=a²-(a*a/4)
9a=a²-(a²/4)

=> (3/4)a² = 9a
=>(3/4)a=9
=>a=12

Therefore the three sides of the triangle are:
a-d=9

a=12

a+d=15

We can check that these are truly Pythagorean triples since

9²+12²=15²

The question asked for only b though so

b=12

Timijo:
Thank you for the correction. Please, I will need to keep in touch with you. I need help with some questions. Please send me your WhatsApp number via email. I have sent you an email.

I can't receive emails through Nairaland for some reason

I'll send you one

Najdorf:

I can't receive emails through Nairaland for some reason

I'll send you one

I have replied you, please check.

Bros, please I need your help ooo. I'm stuck with these questions. Can you send me your WhatsApp number via email?

Najdorf:

I can't receive emails through Nairaland for some reason

I'll send you one

Timijo:
Bros, please I need your help ooo. I'm stuck with these questions. Can you send me your WhatsApp number via email?

.