Question

determine the lengths of the legs of each 45°-45°-90° triangle. write your answer as a radical.
9
16 cm

a

a√2 = 16

a = 16/√2

a = 8√2 cm
10
12 mi

a

a√2 = 12

a = 12/√2
11
6√2 ft

a

a
12
8√2 m

a

a
use the given information to answer each question. round your answer to the nearest tenth, if necessary.
13 soren is flying a kite on the beach. the string forms a 45° angle with the ground. if he has let out 16 meters of line, how high above the ground is the kite?
the height of the kite above the ground is the leg of a 45°-45°-90° right triangle, and the length of the line is the hypotenuse. if the hypotenuse is 16 meters, the leg is 8√2 meters. the kite is approximately 11.3 meters above the ground.
14 meena is picking oranges from the tree in her yard. she rests a 12-foot ladder against the tree at a 45° angle. how far is the top of the ladder from the ground?

Explanation:

Problem 9:

Step1: Recall 45-45-90 triangle ratio

In a \(45^\circ - 45^\circ - 90^\circ\) triangle, hypotenuse \(= a\sqrt{2}\), where \(a\) is leg length. Given hypotenuse \(= 16\) cm, so \(a\sqrt{2}=16\).

Step2: Solve for \(a\)

\(a=\frac{16}{\sqrt{2}}\), rationalize denominator: \(a = \frac{16\sqrt{2}}{2}=8\sqrt{2}\) cm.

Step1: Use 45-45-90 triangle formula

Hypotenuse \(= a\sqrt{2}\), hypotenuse \(= 12\) mi, so \(a\sqrt{2}=12\).

Step2: Find \(a\)

\(a=\frac{12}{\sqrt{2}}=\frac{12\sqrt{2}}{2}=6\sqrt{2}\) mi.

Step1: Apply 45-45-90 triangle ratio

Hypotenuse \(= a\sqrt{2}\), hypotenuse \(= 6\sqrt{2}\) ft, so \(a\sqrt{2}=6\sqrt{2}\).

Step2: Solve for \(a\)

Divide both sides by \(\sqrt{2}\): \(a = 6\) ft.

Answer:

\(8\sqrt{2}\) cm

Problem 10: