Find the Hypotenuse Length in a Right Triangle with Perimeter 12+4√5 cm

Question

Look at the triangle in the figure.

What is the length of the hypotenuse given that its perimeter is 12+45 12+4\sqrt{5} cm?

444AAABBBCCC

Video Solution

Solution Steps

00:00 Find the length of the cathetus
00:03 The perimeter of the triangle equals the sum of its sides
00:12 Substitute in the relevant values and calculate to express BC
00:23 Isolate BC
00:29 This is the expression of BC using CA
00:37 Apply the Pythagorean theorem
00:43 Substitute in the relevant values according to our given data and calculation
00:59 Expand the parentheses for AC
01:07 Square AC
01:14 Proceed to expand the second set of parentheses so that we can reduce
01:44 Move AC to one side of the equation
02:00 Remove AC from the parentheses
02:15 Remove 8 from the parentheses
02:23 Reduce wherever possible and we should obtain the size of AC
02:27 Substitute the size of AC into the expression for the side BC
02:37 This is the solution
02:37 This is the solution

Step-by-Step Solution

We calculate the perimeter of the triangle:

12+45=4+AC+BC 12+4\sqrt{5}=4+AC+BC

As we want to find the hypotenuse (BC), we isolate it:

12+454AC=BC 12+4\sqrt{5}-4-AC=BC

BC=8+45AC BC=8+4\sqrt{5}-AC

Then calculate AC using the Pythagorean theorem:

AB2+AC2=BC2 AB^2+AC^2=BC^2

42+AC2=(8+45AC)2 4^2+AC^2=(8+4\sqrt{5}-AC)^2

16+AC2=(8+45)22×AC(8+45)+AC2 16+AC^2=(8+4\sqrt{5})^2-2\times AC(8+4\sqrt{5})+AC^2

We then simplify the two:AC2 AC^2

16=82+2×8×45+(45)22×8×AC2AC45 16=8^2+2\times8\times4\sqrt{5}+(4\sqrt{5})^2-2\times8\times AC-2AC4\sqrt{5}

16=64+645+16×516AC85AC 16=64+64\sqrt{5}+16\times5-16AC-8\sqrt{5}AC

16AC+85AC=64+645+16×516 16AC+8\sqrt{5}AC=64+64\sqrt{5}+16\times5-16

AC(16+85)=128+645 AC(16+8\sqrt{5})=128+64\sqrt{5}

AC=128+64516+85=8(16+85)16+85 AC=\frac{128+64\sqrt{5}}{16+8\sqrt{5}}=\frac{8(16+8\sqrt{5})}{16+8\sqrt{5}}

We simplify to obtain:

AC=8 AC=8

Now we can replace AC with the value we found for BC:

BC=8+45AC BC=8+4\sqrt{5}-AC

BC=8+458=45 BC=8+4\sqrt{5}-8=4\sqrt{5}

Answer

45 4\sqrt{5} cm