Before you read

Answer first — see what you already know.

A square tile is cut diagonally, creating a 45-45-90 triangle. If each leg of the tile measures 7 inches, what is the diagonal (hypotenuse)?

A
7
B
$72$
C
$73$
D
14

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A construction worker measures a square beam: its diagonal is 12 cm. What is the side length?

A
6 cm
B
$62$ cm
C
$63$ cm
D
$122$ cm

Check your answer

Answer:

Theory

The Two Special Triangles

45-45-90 Triangle (isosceles right triangle):
Sides in ratio $1 : 1 : \sqrt{2}$
If legs $= a$ , hypotenuse $= a\sqrt{2}$

45-45-90: sides in ratio 1:1:√2

30-60-90 Triangle:
Sides in ratio $1 : \sqrt{3} : 2$
- Side opposite $30°$ = $x$ (shortest)
- Side opposite $60°$ = $x\sqrt{3}$
- Side opposite $90°$ = $2x$ (hypotenuse)

30-60-90: sides in ratio 1:√3:2

Example 1

A 30-60-90 triangle has hypotenuse 10. Find the other sides.

$2x = 10$ , so $x = 5$

Short leg (opposite 30 $^{\circ}$ ) = 5

Long leg (opposite 60 $^{\circ}$ ) = $5\sqrt{3}$

5 and

5\sqrt{3}

Strategy

SAT Pro Tip

These ratios are on the SAT reference sheet, but memorizing them saves precious time. Quick trick: in a 30-60-90, the hypotenuse is always DOUBLE the short leg. In a 45-45-90, multiply a leg by $\sqrt{2}$ to get the hypotenuse.

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