Respuesta :

msm555

Answer:

[tex] a = 2\sqrt{3} [/tex]

[tex] h = 2\sqrt{2} [/tex]

Step-by-step explanation:

For the left triangle:

Given:

  • Angle: [tex]30^\circ[/tex]
  • Adjacent side: [tex]3[/tex]
  • Hypotenuse: [tex]a[/tex]

We can use the cosine of the angle to relate the adjacent side and the hypotenuse:

[tex] \cos(30^\circ) = \dfrac{\textsf{Adjacent}}{\textsf{Hypotenuse}} [/tex]

[tex] \cos(30^\circ) = \dfrac{3}{a} [/tex]

The cosine of [tex]30^\circ[/tex] is [tex] \dfrac{\sqrt{3}}{2} [/tex], so:

[tex] \dfrac{\sqrt{3}}{2} = \dfrac{3}{a} [/tex]

Now, solve for [tex]a[/tex]:

[tex] a = \dfrac{3}{\dfrac{\sqrt{3}}{2}} [/tex]

[tex] a = \dfrac{3 \times 2}{\sqrt{3}} [/tex]

[tex] a = \dfrac{6}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} [/tex]

[tex] a = \dfrac{6\sqrt{3}}{3} [/tex]

[tex] a = 2\sqrt{3} [/tex]

So, the value of [tex]a[/tex] for the left triangle is [tex]2\sqrt{3}[/tex].

For the right triangle:

Given:

  • Given: Angle: [tex]45^\circ[/tex]
  • Opposite side: [tex]2[/tex]
  • Hypotenuse: [tex]h[/tex]

We can use the sine of the angle to relate the opposite side and the hypotenuse:

[tex] \sin(45^\circ) = \dfrac{\textsf{Opposite}}{\textsf{Hypotenuse}} [/tex]

[tex] \sin(45^\circ) = \dfrac{2}{h} [/tex]

The sine of [tex]45^\circ[/tex] is [tex] \dfrac{\sqrt{2}}{2} [/tex], so:

[tex] \dfrac{\sqrt{2}}{2} = \dfrac{2}{h} [/tex]

Now, solve for [tex]h[/tex]:

[tex] h = \dfrac{2}{\dfrac{\sqrt{2}}{2}} [/tex]

[tex] h = \dfrac{2 \times 2}{\sqrt{2}} [/tex]

[tex] h = \dfrac{4}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} [/tex]

[tex] h = \dfrac{4\sqrt{2}}{2} [/tex]

[tex] h = 2\sqrt{2} [/tex]

So, the value of [tex]h[/tex] for the right triangle is [tex]2\sqrt{2}[/tex].

semsee45

Answer:

[tex]a=\boxed{2\sqrt{3}}[/tex]

[tex]h = \boxed{2\sqrt{2}}[/tex]

Step-by-step explanation:

30-60-90 Right Triangle

The first triangle is a special 30-60-90 right triangle because its interior angles measure 30°, 60° and 90°.

In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where the hypotenuse is twice the length of the shortest leg.

In this triangle, the longest leg measures 3 units.

Therefore, to find the length of the hypotenuse (a), divide the length of the longest leg by √3, then multiply it by 2:

[tex]a=\dfrac{3}{\sqrt{3}} \cdot 2 \\\\\\\\ a=\dfrac{6}{\sqrt{3}} \\\\\\\\ a=\dfrac{6\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}} \\\\\\\\ a=\dfrac{6\cdot \sqrt{3}}{3} \\\\\\\\ a=2\sqrt{3}[/tex]

Therefore, the exact value of the side length a is:

[tex]\LARGE\boxed{\boxed{a=2\sqrt{3}}}[/tex]

[tex]\dotfill[/tex]

45-45-90 Right Triangle

The second triangle is a special 45-45-90 right triangle because its interior angles measure 45°, 45° and 90°.

In a 45-45-90 triangle, the sides are in the ratio 1 : 1 : √2, where the two legs are equal in length, and the hypotenuse is √2 times the length of one of the legs.

In this triangle, the length of a leg 2 units.

Therefore, to find the length of the hypotenuse (h), simply multiply the length of the leg by √2:

[tex]h = 2\sqrt{2}[/tex]

Therefore, the exact value of the side length h is:

[tex]\LARGE\boxed{\boxed{h=2\sqrt{2}}}[/tex]

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