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Sprnt

Answer:

The length of side NH is 11.66.

The length of side GX is 4.89.

The length of side GB is 3.86.

Step-by-step explanation:

Check the attached image below to better understand the method used for the solutions.

1. Length of side NH.

Using the pythagoream theorem, length of side NH is:

[tex](19)^2=(15)^2+(NH)^2\\ \\(NH)^2=(19)^2-(15)^2\\ \\NH=\sqrt{(19)^2-(15)^2} \\ \\NH=\sqrt{361-225} \\ \\NH=\sqrt{136} \\\\NH=11.66[/tex]

2. Length of side GX.

Considering that these triangles are similar, the ratios between their sides are the same. Hence:

[tex]\frac{GX}{BX} =\frac{UH}{NH} \\ \\\frac{GX}{3} =\frac{19}{11.66 } \\ \\GX=\frac{19}{11.66 } *3\\\\ GX=4.89[/tex]

3. Length of side GB.

Use the same logic from the previous solution.

[tex]\frac{GB}{BX} =\frac{UN}{NH} }\\ \\\frac{GB}{3} =\frac{15}{11.66 }\\ \\GB=\frac{15}{11.66 }*3\\ \\GB=3.86[/tex]

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semsee45

Answer:

[tex]\sf NH=2\sqrt{34} \approx 11.7 \:\:(nearest\:tenth)[/tex]

[tex]\sf GX=\dfrac{57\sqrt{34}}{68} \approx 4.9 \:\:(nearest\:tenth)[/tex]

[tex]\sf GB=\dfrac{45\sqrt{34}}{68} \approx 3.9 \:\:(nearest\:tenth)[/tex]

Step-by-step explanation:

Similar Triangles

Two triangles are similar if their corresponding angles are the same size.

In similar triangles, corresponding sides are always in the same ratio.

From inspection of the given diagram ΔUNH and ΔGBX are similar as their corresponding angles are the same size.

As the length of two sides of ΔUNH have been given, find NH by using Pythagoras Theorem.

Pythagoras Theorem

[tex]a^2+b^2=c^2[/tex]

where:

  • a and b are the legs of the right triangle.
  • c is the hypotenuse (longest side) of the right triangle.

Therefore:

[tex]\implies \sf NH^2+UN^2=UH^2[/tex]

[tex]\implies \sf NH^2+15^2=19^2[/tex]

[tex]\implies \sf NH^2+225=361[/tex]

[tex]\implies \sf NH^2=136[/tex]

[tex]\implies \sf NH=\sqrt{136}[/tex]

[tex]\implies \sf NH=\sqrt{4 \cdot 34}[/tex]

[tex]\implies \sf NH=\sqrt{4}\sqrt{34}[/tex]

[tex]\implies \sf NH=2\sqrt{34}[/tex]

As corresponding sides are always in the same ratio in similar triangles:

[tex]\sf \implies GB :UN = BX:NH = GX:UH[/tex]

[tex]\sf \implies GB:15 = 3:2\sqrt{34} = GX:19[/tex]

[tex]\implies \sf \dfrac{GB}{15}=\dfrac{3}{2\sqrt{34}}=\dfrac{GX}{19}[/tex]

Length of side GX:

[tex]\implies \sf \dfrac{3}{2\sqrt{34}}=\dfrac{GX}{19}[/tex]

[tex]\implies \sf GX=\dfrac{3 \cdot 19}{2\sqrt{34}}[/tex]

[tex]\implies \sf GX=\dfrac{57}{2\sqrt{34}}[/tex]

[tex]\implies \sf GX=\dfrac{57\sqrt{34}}{68}[/tex]

Length of side GB:

[tex]\implies \sf \dfrac{GB}{15}=\dfrac{3}{2\sqrt{34}}[/tex]

[tex]\implies \sf GB=\dfrac{15 \cdot 3}{2\sqrt{34}}[/tex]

[tex]\implies \sf GB=\dfrac{45}{2\sqrt{34}}[/tex]

[tex]\implies \sf GB=\dfrac{45\sqrt{34}}{68}[/tex]

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