Question

Pls help thx (dont guess pls)

Answer 1

UV = 10, ∠ UCT = 44° , ∠ UAV = 20°

Explanation:

• the incentre (V) is equally distant from the triangles three sides

Then

UV = SV = 10

The angle bisector VC , splits ∠ UCT into 2 equal parts

Then

∠ UCT = 2 × ∠ UCV = 2 × 22° = 44°

the sum of the 3 angles in Δ ABC = 180° , that is

∠ SAU + ∠ SBT + ∠ UCT = 180° ( substitute values )

∠ SAU + 96° + 44° = 180°

∠ SAU + 140° = 180° ( subtract 140° from both sides )

∠ SAU = 40°

Since VA bisects ∠ SAU

Then

∠ UAV =
`(1)/(2)` × ∠ SAU =
`(1)/(2)` × 40° = 20°

Answer 2

1. UV:
`\(UV \approx 10 + 13 \cdot \tan(84^\circ) + 0.92 \cdot \tan(22^\circ) \cdot (\tan(42^\circ) \cdot (s - 13))\).`

2.
`\(m\angle UCT\): \(m\angle UCT \approx 54^\circ\)`.

3.
`\(m\angle UAV\): \(m\angle UAV = 84^\circ\)`.


Let's denote the angles as follows:

`- \(\angle AVB = \alpha\),- \(\angle AVC = \beta\),- \(\angle BVC = \gamma\).`

We are given that
`\(m\angle SBT = 96^\circ\) and \(m\angle UCV = 22^\circ\). Since \(BT\) and \(UC\) are angle bisectors, we can find that \(\angle SBU = \angle UBV = \alpha/2\) and \(\angle UCB = \angle VCU = \beta/2\).`

Now, let's find
`\(\angle SBV\)`:

`\[m\angle SBV = m\angle SBU + m\angle UBV = (\alpha)/(2) + (\alpha)/(2) = \alpha.\]`

Similarly, let's find
`\(\angle UCV\)`:

`\[m\angle UCV = m\angle UCB + m\angle VCU = (\beta)/(2) + (\beta)/(2) = \beta.\]`

Now, we have the following information:

`- \(m\angle SBV = \alpha\),- \(m\angle UCV = \beta\),- \(m\angle SBT = 96^\circ\),- \(m\angle UCV = 22^\circ\).`

Since
`\(\angle SBV + \angle UCV + \angle SBT = 180^\circ\)` in a triangle, we can write the equation:

`\[\alpha + \beta + 96^\circ = 180^\circ.\]`

Solving for
`\(\alpha + \beta\)`:

`\[\alpha + \beta = 180^\circ - 96^\circ = 84^\circ.\]`

Now, let's find
`\(\angle UBV\)` and
`\(\angle UCB\)` using the fact that they are half of
`\(\alpha\)` and
`\(\beta\)`, respectively:

`\[\angle UBV = (\alpha)/(2) = (84^\circ)/(2) = 42^\circ,\]\[\angle UCB = (\beta)/(2) = (84^\circ)/(2) = 42^\circ.\]`

Finally, we have:

-
`\(UV = SV + SU = SV + BV \tan(\angle SBV) + CV \tan(\angle UCV),\)`

-
`\(m\angle UCT = 180^\circ - \angle UCB - \angle UCV,\)`

-
`\(m\angle UAV = \angle UBV + \angle UCB.\)`

1. UV:

`\[UV = 10 + 13 \tan(84^\circ) + \tan(22^\circ) \cdot (\tan(42^\circ) \cdot (s - 13)).\]`

First, calculate
`\(s\)` and
`\(a\)`:

`\[s = (13 + CV + BC) / 2\]`

`\[a = 13.\]`

Substitute these values into the expression for UV:

`\[UV = 10 + 13 \tan(84^\circ) + \tan(22^\circ) \cdot (\tan(42^\circ) \cdot (s - 13)).\]`

Using a calculator, we find that
`\(UV \approx 10 + 13 \cdot \tan(84^\circ) + \tan(22^\circ) \cdot (\tan(42^\circ) \cdot (s - 13)) \approx 10 + 13 \cdot 11.43 + 0.92 \cdot (\tan(42^\circ) \cdot (s - 13))\).`

2.
`\(m\angle UCT\)`:

`\[m\angle UCT = 180^\circ - 42^\circ - 84^\circ.\]`

Therefore,
`\(m\angle UCT = 180^\circ - 42^\circ - 84^\circ \approx 54^\circ.\)`

3.
`\(m\angle UAV\)`:

`\[m\angle UAV = 42^\circ + 42^\circ.\]`

Therefore,
`\(m\angle UAV = 42^\circ + 42^\circ = 84^\circ.\)`