identifying a Leg Length.

Question

what is the length of PR? ______units.

PR is 5n. QR is 32+n

 

Answer 1

Hi Lisa,

 

The congruent sides of an isosceles triangle are called its legs - the same term as for the two sides of a right triangle that form the 90⁰ angle.

 

Case 1:  If I have an isosceles triangle, PQR, and PR is a leg of length 5n, then if PQ is the other leg, PQ has length 5n.

 

Case 2:  If my isosceles triangle has one leg QR with length 5n + 32, the leg PQ is 5n+ 32, but without an angle measure to help, finding PR length is trickier.

 

Case 3:  The two congruent legs do form a right angle. since n must be a positive number for 5n to represent a length, PR is a leg, and QR is the hypotenuse, then the length of PQ is 5n.

 

    Using the 45-45-90⁰ ratio rule, the hypotenuse is 5n*(SQRT2). So 5n*(SQRT2) = 5n + 32, solving for 5n yields 5n= 32/(SQRT2 -1) or approx 77.255 units

 

 

Using the Pythagorean Theorem,

                             (5n)^2 + (5n)^2 = (5n + 32)^2

 

                               25n² + 25n²= 25n² + 320n + 1024 = 0

 

                               25n² - 320n - 1024 = 0

 

                               a = 25, b = -320, c = -1024

 

                                n = (320 + SQRT204800)/50.

 

                             Using simplified exact result with radical, 5n = (320 + 320*SQRT2)/10 units, or approx 77.255 units