Question 25·Easy·Right Triangles and Trigonometry
In a right triangle, the length of the hypotenuse is 13 centimeters, and the length of one leg is 5 centimeters.
What is the length, in centimeters, of the triangle's other leg?
For right triangle side-length questions, immediately decide which side is the hypotenuse and apply the Pythagorean theorem . If you are given the hypotenuse and one leg, plug in those values, subtract the square of the known leg from the square of the hypotenuse, and then take the positive square root. To save time, memorize common Pythagorean triples (such as 3-4-5, 5-12-13), which can let you spot the missing side without doing full calculations.
Hints
Recall the key relationship
In any right triangle, how are the lengths of the two legs and the hypotenuse related? Think of the Pythagorean theorem.
Write an equation
Let be the length of the unknown leg. Use the equation with and one leg equal to 5 to set up an equation involving .
Isolate the unknown
After substituting, move the known squared value to the other side of the equation to find . Then consider what you must do to get itself.
Desmos Guide
Use Desmos to compute the missing side
In Desmos, type sqrt(13^2 - 5^2) and press Enter. The value that Desmos outputs is the length of the triangle's other leg.
Step-by-step Explanation
Identify the sides and the theorem
In a right triangle, the side opposite the right angle is the hypotenuse. Here, the hypotenuse is 13 cm, and one leg is 5 cm. The relationship between the sides is given by the Pythagorean theorem:
where is the hypotenuse and and are the legs.
Set up the equation for the unknown leg
Let be the length of the unknown leg. Substitute the known values into the Pythagorean theorem:
Now simplify the squares:
Solve for the square of the unknown leg
Isolate by subtracting 25 from both sides:
Compute the difference:
Find the length of the unknown leg
Take the positive square root of both sides (lengths are positive):
So the length of the other leg is centimeters.