In this rectangle, 𝑋𝑍 equals seven 𝑐, 𝑍𝑌 equals 21, and 𝑋𝑌 equals 28. Find the value of 𝑐.
The diagonal of our rectangle 𝑋𝑍 is equal to seven 𝑐, 𝑍𝑌 is equal to 21, and 𝑋𝑌 is equal to 28. These three sides create a right-angled triangle as the angles in the corner of a rectangle are 90 degrees. In any right-angled triangle. We can use the Pythagorean theorem to calculate the length of a missing side. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse.
Substituting in our values, we have 21 squared plus 28 squared is equal to seven 𝑐 squared. 21 squared is equal to 441; 28 squared is 784. Squaring seven 𝑐 means multiplying seven 𝑐 by itself. This gives us 49𝑐 squared as seven squared is 49 and 𝑐 multiplied by 𝑐 is 𝑐 squared.
The left-hand side of the equation simplifies to 1225. We can then divide both sides of this equation by 49 to calculate a value of 𝑐 squared. 1225 divided by 49 is 25. Finally, square rooting both sides of this equation gives us a value of 𝑐 equal to five.
An alternative and slightly quicker way to solve this problem would be to recall one of our Pythagorean triples. The Pythagorean triple three, four, five means that any triangle whose sides are in the ratio three, four, five will be right angled. Three multiplied by seven is equal to 21, which is the length of 𝑌𝑍. Four multiplied by seven is equal to 28, which is the length of 𝑋𝑌. Multiplying five by seven gives us 35. So this must be the length of 𝑋𝑍. This means that seven 𝑐 is equal to 35. Dividing both sides by seven, we once again get the answer 𝑐 is equal to five.