Turn on thread page Beta
    • Thread Starter
    Offline

    16
    ReputationRep:
    An isosceles right-angled triangle has its two shorter sides of length a. Write down an expression for its perimeter in terms of a.

    A length of rope 10 meters long is to be pegged out to form an isosceles right-angled triangle. Find in as simple a form as possible, exact expressions for the lengths of the sides.

    I'm totally lost at this question. I've got the answers in the back of the book but no idea of how to get there. Can somebody explain the process fully? Thanks
    Offline

    10
    ReputationRep:
    "isosceles right-angled triangle"...so you know exactly what the angle are.
    "two shorter sides of length a"...so you know which sides they must be, can you work out the other side using pythagoras?
    Offline

    5
    ReputationRep:
    (Original post by Bleak Lemming)
    An isosceles right-angled triangle has its two shorter sides of length a. Write down an expression for its perimeter in terms of a.

    A length of rope 10 meters long is to be pegged out to form an isosceles right-angled triangle. Find in as simple a form as possible, exact expressions for the lengths of the sides.

    I'm totally lost at this question. I've got the answers in the back of the book but no idea of how to get there. Can somebody explain the process fully? Thanks
    For the first part, you need to work out the length of the hypotenuse. The two shorter sides are both length a, so by using Pythagoras' theorem the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. This gives C² = a² + a² where C is the length of the hypotenuse. Square rooting both sides will give C = sqrt(2a²) = sqrt(2)*a. Adding all 3 sides together will give the perimeter.

    For the second part, you just need equate this to 10 and then solve for a, and substitute this in to the expression for the hypotenuse.
    • Thread Starter
    Offline

    16
    ReputationRep:
    (Original post by Phil_Waite)
    For the first part, you need to work out the length of the hypotenuse. The two shorter sides are both length a, so by using Pythagoras' theorem the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. This gives C² = a² + a² where C is the length of the hypotenuse. Square rooting both sides will give C = sqrt(2a²) = sqrt(2)*a. Adding all 3 sides together will give the perimeter.

    For the second part, you just need equate this to 10 and then solve for a, and substitute this in to the expression for the hypotenuse.
    Yea the first part is

    a(2+\sqrt2)

    I just don't get where \sqrt2 came from...
    Offline

    15
    ReputationRep:
    (Original post by Bleak Lemming)
    Yea the first part is

    a(2+\sqrt2)

    I just don't get where \sqrt2 came from...
    It's because the two shorter sides cannot be the hypoteneuse (otherwise the triangle does not exist) and thus the length of the hypoteneuse (say, C) is given by (as it is a right-angled triangle):

    C^2 = a^2 + a^2 => C = \sqrt{2a^2}=a \sqrt2

    Then clearly the 2a comes from the other two sides.
    • Thread Starter
    Offline

    16
    ReputationRep:
    (Original post by marcusmerehay)
    It's because the two shorter sides cannot be the hypoteneuse (otherwise the triangle does not exist) and thus the length of the hypoteneuse (say, C) is given by (as it is a right-angled triangle):

    C^2 = a^2 + a^2 => C = \sqrt{2a^2}=a \sqrt2
    Got it, thanks :]
    Offline

    10
    ReputationRep:
    You should get the perimeter as a + a + sqrt(a^2) = 2a + sqrt(2)a = a( 2 + sqrt(2)).
    Offline

    13
    ReputationRep:
    1) To Find the hypotenuse, since its a right angled triangle and isoceles, 2 sides will be of length a, and we find the hypotenuse.
    Hypotenuse  = \sqrt{a^2+a^2}

= \sqrt {2a^2}

= \sqrt 2a.

    Perimeter
     =  a+a+\sqrt 2a

= 2a+\sqrt 2a = a(2+\sqrt 2)

    2) Since they say that the perimeter is now 10 metres,
    10  = a(2+\sqrt 2)

    \small a= \frac {10}{\sqrt 2 + 2}

    = 5(2-\sqrt 2)

    Therefore you know the two sides, and the hypotenuse is then
    Hypotenuse  = 10\sqrt2 - 10
    Offline

    5
    ReputationRep:
    (Original post by Bleak Lemming)
    Yea the first part is

    a(2+\sqrt2)

    I just don't get where \sqrt2 came from...
    Do you understand the part where the square of the hypotenuse is equal to the sum of the squares of both sides? C² = a²+a² where C is the length of the hypotenuse. If you take the square root of both sides, you get:
    C = \sqrt{a^2+a^2} = \sqrt{2a^2} = \sqrt{2}a

    Adding this to the length of the other two sides:

    \sqrt{2}a + a + a = (\sqrt{2}+2)a

    Edit - Too late
    Offline

    13
    ReputationRep:
    If anybody also knows LaTex properly can you please help me to mend the 10/2+sqrt2 part
    Offline

    5
    ReputationRep:
    a= \frac{10}{2+\sqrt 2}
    Offline

    13
    ReputationRep:
    Thats what I wrote. I don't seem to understand why it looks like that
    Offline

    13
    ReputationRep:
    \small a= \frac {10}{\sqrt 2 + 2}
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 31, 2011

University open days

  1. University of Cambridge
    Christ's College Undergraduate
    Wed, 26 Sep '18
  2. Norwich University of the Arts
    Undergraduate Open Days Undergraduate
    Fri, 28 Sep '18
  3. Edge Hill University
    Faculty of Health and Social Care Undergraduate
    Sat, 29 Sep '18
Poll
Which accompaniment is best?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.