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Difficulty: MediumTriangles and Pythagorean Theorem

A custom corner desk is shaped like a right triangle so it fits perfectly against two perpendicular walls. The sections of the desk that touch the walls measure 24 inches24\text{ inches} and 10 inches10\text{ inches} in length. What is the length of the front edge of the desk in inches?

Answer: 26 inches

Answer

26
Because the walls meet at a perpendicular right angle, the corner desk forms a right triangle. The edges touching the walls act as the legs (a=24a = 24, b=10b = 10), and the front edge is the hypotenuse (cc). Using the Pythagorean Theorem (a2+b2=c2a^2 + b^2 = c^2), we square both legs to get 576576 and 100100. Adding these together gives 676676. The square root of 676676 is 2626, which is the exact length of the front edge in inches.

Step-by-Step Solution

1
Identify the known sides of the right triangle
Legs are a=24a = 24 and b=10b = 10.
The walls are perpendicular, forming a 90∘90^{\circ} angle. Therefore, the edges touching the walls are the legs of a right triangle.
2
Set up the Pythagorean Theorem
242+102=c224^2 + 10^2 = c^2
The Pythagorean Theorem (a2+b2=c2a^2 + b^2 = c^2) is the standard formula used to find the missing hypotenuse (cc) when both legs are known.
3
Square the lengths of the legs
576+100=c2576 + 100 = c^2
Evaluating the exponents is the required next step in the order of operations.
4
Add the squared values
676=c2676 = c^2
Combining the squares of the legs gives the square of the hypotenuse.
5
Find the square root of the sum
c=26c = 26
Taking the square root of 676676 isolates cc, revealing the true length of the desk's front edge.

Key Concept

Using the Pythagorean Theorem (a2+b2=c2a^2 + b^2 = c^2) to calculate the length of a hypotenuse.

Alternative Method

You can solve this rapidly without large calculations by recognizing special Pythagorean triples. This triangle is a scaled-up version of the fundamental 5−12−135-12-13 right triangle. Multiplying each side of a 5−12−135-12-13 triangle by 22 gives side lengths of 1010, 2424, and 2626. Since the legs are 1010 and 2424, the hypotenuse must be 2626.
Estimated Time:45s
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Topics

Triangles and Pythagorean Theorem

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