Answer :

olumideolawoyin

Answer:

[tex]b = \frac{45}{4} [/tex]

Step-by-step explanation:

Reference to angle 30⁰ and using SOH CAH TOA,

cos 30⁰ = adjacent / hypotenuse

cos 30⁰ = b / z

[tex] \frac{ \sqrt{3} }{2} = \frac{b}{z} [/tex]

z√3 = 2b Divide both sides by √3

[tex]z = \frac{2b}{ \sqrt{3} } [/tex]

When rationalized,

[tex]z = \frac{2b \sqrt{3} }{3} [/tex]

Also considering angle 60⁰ and using SOH CAH TOA,

[tex]sin 60⁰ = \frac{z}{15} [/tex]

z = 15 × sin60⁰

[tex]z = 15 \times \frac{ \sqrt{3} }{2} [/tex]

Z has two values then we can equate the two values.

[tex] \frac{15 \sqrt{3} }{2} = \frac{2b \sqrt{3} }{3} [/tex]

Multiply both sides by 6

3 × 15√3 = 2 × 2b√3

45√3 = 4b√3

Divide both sides by 4√3

[tex]b = \frac{45 \sqrt{3} }{4 \sqrt{3} } [/tex]

Therefore,

[tex]b = \frac{45}{4} [/tex]

semsee45

Answer:

[tex]b = \dfrac{45}{4}[/tex]

Step-by-step explanation:

The given diagram shows a 30-60-90 right triangle where:

  • x is the leg opposite the 30° angle.
  • z is the leg opposite the 60° angle.
  • The hypotenuse (a + b) measures 15 units.

In a 30-60-90 right triangle, the sides are in the ratio 1 : √3 : 2. This means the hypotenuse is twice the length of the leg opposite the 30° angle.

In this triangle, the hypotenuse measures 15 units, so the length of side z opposite the 60° angle is √3/2 times the hypotenuse:

[tex]z= \dfrac{15\sqrt{3}}{2}[/tex]

An altitude (y) is drawn from the right angle of the 30-60-90 triangle to its hypotenuse, dividing the triangle into two smaller similar 30-60-90 triangles.

In the rightmost of these smaller triangles, the hypotenuse is side z and the longest leg is side b. Applying the ratio 1 : √3 : 2, the length of side b is √3/2 times side z:

[tex]b = \dfrac{15\sqrt{3}}{2}\cdot \dfrac{\sqrt{3}}{2}\\\\\\b=\dfrac{15 \cdot 3}{4}\\\\\\b=\dfrac{45}{4}[/tex]

Therefore, the value of b is:

[tex]\Large\boxed{\boxed{b=\dfrac{45}{4}}}[/tex]