Question

1. the legs of an isosceles right - triangle are each 8 inches long. what is the length of the hypotenuse? a. 24 inches b. 16 inches c. 8√5 inches d. 8√2 inches 5. the shorter leg of a 30 - 60 - 90 triangle measures 7 inches. what is the length of the longer leg? a. 7√2 inches b. 7√3 inches c. 14 inches d. 21 inches 6. a kite is attached to the ground with a 55 - foot string. it is flying at a vertical height of 30 feet. what is the approximate angle the string makes with the ground? a. 29° b. 33° c. 57° d. 61° 7. a building is 75 meters tall and casts a shadow. the angle of elevation from the tip of the shadow to the sun is 48°. what is the approximate length of the shadow cast by the building? a. 50.2° b. 55.7° c. 67.5° d. 83.3°

Explanation:

Step1: Recall Pythagorean theorem for isosceles right - triangle

For an isosceles right - triangle with leg length $a = 8$ inches, by the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, and since $a = b = 8$, we have $c^{2}=8^{2}+8^{2}$.

Step2: Calculate the hypotenuse

$c^{2}=64 + 64=128$, so $c=\sqrt{128}=8\sqrt{2}$ inches.

Step3: Recall side - length relationship for 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the shorter leg (opposite 30°) has length $a = 7$ inches, the longer leg (opposite 60°) has length $b=a\sqrt{3}=7\sqrt{3}$ inches.

Step4: Use trigonometry for kite problem

Let the angle the string makes with the ground be $\theta$. We know the vertical height (opposite side) $y = 30$ feet and the length of the string (hypotenuse) $r = 55$ feet. Using $\sin\theta=\frac{y}{r}$, so $\sin\theta=\frac{30}{55}\approx0.5455$. Then $\theta=\sin^{- 1}(0.5455)\approx33^{\circ}$.

Step5: Use trigonometry for building - shadow problem

Let the height of the building $h = 75$ meters be the opposite side and the length of the shadow $x$ be the adjacent side. The angle of elevation $\theta = 48^{\circ}$. Using $\tan\theta=\frac{h}{x}$, so $x=\frac{h}{\tan\theta}=\frac{75}{\tan48^{\circ}}\approx75\div1.1106\approx67.5$ meters. But the options seem to be angles which is a mistake in the problem setup. If we assume we want to find the angle of depression from the top of the building to the tip of the shadow, it is the complementary angle of the angle of elevation.

Answer:

1. D. $8\sqrt{2}$ inches
2. B. $7\sqrt{3}$ inches
3. B. $33^{\circ}$
4. (There is an error in the options as the problem asks for length of shadow but options are angles. If we assume correct problem setup, the length of shadow is approximately 67.5 meters)