QuanticaPrealgebraOpen in the courseCut a square in half along its diagonal and you get a right triangle with two equal legs. Triangles like this show up everywhere, in ramps, rooftops, and folded napkins, so it pays to know them by heart. Start with the smallest one and measure its hypotenuse with the Pythagorean Theorem.
Problem
A right triangle has legs \(1\) and \(1\). Use the Pythagorean Theorem to find the square of the hypotenuse. What number is \(c^2\)?
Show a hint
- The theorem says \(a^2 + b^2 = c^2\), and here both legs are \(1\).
- Compute \(1^2 + 1^2\).
Show the full solution
The legs give \(c^2 = 1^2 + 1^2 = 2\), so the hypotenuse itself is \(\sqrt{2}\). The square of the hypotenuse is \(\boxed{2}\).
Problem
Each leg of a 45-45-90 triangle is \(5\). The hypotenuse simplifies to \(k\sqrt{2}\). What whole number is \(k\)?
Show a hint
- The hypotenuse of a 45-45-90 triangle is the leg times \(\sqrt{2}\).
- With legs of \(5\), the hypotenuse is \(5\sqrt{2}\).
Show the full solution
Hypotenuse \(= s\sqrt{2} = 5\sqrt{2}\), so \(k = \boxed{5}\). The theorem confirms it, since \(5^2 + 5^2 = 50\) and \(\sqrt{50} = 5\sqrt{2}\).
Problem
A wheelchair ramp rises at a \(45\) degree angle, so its rise and its horizontal run are the two legs of a 45-45-90 triangle. The run is \(12\) meters. How many meters is the rise?
Show a hint
- In a 45-45-90 triangle the two legs are equal.
- The rise and the run are the two legs.
Show the full solution
The legs of a 45-45-90 triangle are equal, so the rise matches the run at \(\boxed{12}\) meters.
Problem
The hypotenuse of a 45-45-90 triangle measures \(7\sqrt{2}\). How long is each leg?
Show a hint
- Run the rule backwards. The hypotenuse is leg \(\times \sqrt{2}\), so the leg is hypotenuse \(\div \sqrt{2}\).
- If \(s\sqrt{2} = 7\sqrt{2}\), what is \(s\)?
Show the full solution
The hypotenuse is always \(s\sqrt{2}\). Matching \(s\sqrt{2} = 7\sqrt{2}\) gives \(s = \boxed{7}\).
Problem
The short leg of a 30-60-90 triangle is \(4\). The hypotenuse is always twice the short leg. How long is the hypotenuse?
Show a hint
- Hypotenuse \(= 2 \times\) short leg.
Show the full solution
Twice the short leg is \(2 \times 4 = \boxed{8}\).
Problem
The hypotenuse of a 30-60-90 triangle is \(14\). How long is the short leg?
Show a hint
- The hypotenuse is twice the short leg, so run it backwards.
- Half of \(14\).
Show the full solution
The short leg is half the hypotenuse, \(14 \div 2 = \boxed{7}\).
Problem
The short leg of a 30-60-90 triangle is \(5\). The long leg simplifies to \(k\sqrt{3}\). What whole number is \(k\)?
Show a hint
- The long leg is the short leg times \(\sqrt{3}\).
- With a short leg of \(5\), the long leg is \(5\sqrt{3}\).
Show the full solution
Long leg \(=\) short leg \(\times \sqrt{3} = 5\sqrt{3}\). So \(k = \boxed{5}\).
Problem
An equilateral triangle has \(10\) cm sides. Its height simplifies to \(k\sqrt{3}\) cm. What whole number is \(k\)?
Show a hint
- The height cuts the triangle into two 30-60-90 halves, and the short leg of each half is half of \(10\).
- Height \(=\) short leg \(\times \sqrt{3}\), and the short leg is \(5\).
Show the full solution
The height splits the base, so the short leg is \(10 \div 2 = 5\), and the height is the long leg, \(5\sqrt{3}\). So \(k = \boxed{5}\).
Practice these ideas
Practice
Each leg of a 45-45-90 triangle is \(3\). The hypotenuse simplifies to \(k\sqrt{2}\). What is \(k\)?
Show the solution
Hypotenuse \(= 3\sqrt{2}\), so \(k = \boxed{3}\).
Practice
The hypotenuse of a 45-45-90 triangle is \(11\sqrt{2}\). How long is each leg?
Show the solution
From \(s\sqrt{2} = 11\sqrt{2}\) we get \(s = \boxed{11}\).
Practice
A square has \(7\) inch sides. Its diagonal simplifies to \(k\sqrt{2}\) inches. What is \(k\)?
Show the solution
The diagonal is side \(\times \sqrt{2} = 7\sqrt{2}\), so \(k = \boxed{7}\).
Practice
The short leg of a 30-60-90 triangle is \(6\). How long is the hypotenuse?
Show the solution
Hypotenuse \(= 2 \times 6 = \boxed{12}\).
Practice
The hypotenuse of a 30-60-90 triangle is \(20\). How long is the short leg?
Show the solution
Half of \(20\) is \(\boxed{10}\).
Practice
The short leg of a 30-60-90 triangle is \(8\). The long leg simplifies to \(k\sqrt{3}\). What is \(k\)?
Show the solution
The long leg is \(8\sqrt{3}\), so \(k = \boxed{8}\).
Practice
An equilateral triangle has \(12\) cm sides. Its height simplifies to \(k\sqrt{3}\) cm. What is \(k\)?
Show the solution
The short leg is \(12 \div 2 = 6\), so the height is \(6\sqrt{3}\) and \(k = \boxed{6}\).
Practice
Each leg of a 45-45-90 triangle is \(9\). Use the Pythagorean Theorem to check the shortcut. What is the square of the hypotenuse?
Show the solution
\(c^2 = 9^2 + 9^2 = 81 + 81 = \boxed{162}\). The shortcut agrees, since \((9\sqrt{2})^2 = 81 \times 2 = 162\).