Mathematics, 13.01.2021 19:00 itsstaytay200

If G is the incenter of triangle ABC, find each measure

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Answer from: emma3216

Applying the definition of the incenter of a triangle, the measures in triangle ABC where G is the incenter are:

$\mathbf{m \angle ABG = 20^{\circ}}$ $\mathbf{m \angle BCA = 22^{\circ}}$ $\mathbf{m \angle BAC = 118^{\circ}}$ $\mathbf{m \angle BAG = 59^{\circ}}$ DG = 4BE = 10.99BG = 11.7GC = 20.4

Recall the following facts about the incenter of a triangle:

The incenter of a triangle is the point where the three angle bisectors that divides each angle vertex into two intersect each other.

The incenter also, is a point equidistant to all three sides of the triangle at right angle.

Applying the properties above, the measures in the image can be calculated as shown below.

Given:

$m \angle EBG = 20^{\circ}\\\\m \angle ECG = 11^{\circ}\\\\CF = 20\\\\EG = 4\\\\BD = 11$

Find $m \angle ABG$ :

$m \angle ABG = m \angle EBG$ (angle bisector definition)

Substitute

$\mathbf{m \angle ABG = 20^{\circ}}$

Find $m \angle BCA$ :

$m \angle BCA = 2(m \angle ECG)$ (angle bisector definition)

Substitute

$m \angle BCA = 2(11)\\\\\mathbf{m \angle BCA = 22^{\circ}}$

Find $m \angle BAC$ :

$m \angle BAC = 180 - (m \angle CBA + m \angle BCA)$ (sum of triangle definition)

Substitute

$m \angle BAC = 180 - (40 + 22)\\\\\mathbf{m \angle BAC = 118^{\circ}}$

Find $m \angle BAG$ :

$m \angle BAG = \frac{1}{2} (m \angle BAC)$ (angle bisector definition)

Substitute

$m \angle BAG = \frac{1}{2} (118)\\\\\mathbf{m \angle BAG = 59^{\circ}}$

Find DG:

DG = EG (Perpendicular bisectors of the sides of a the triangle are equal in length from the incenter of a triangle)

Substitute

$\mathbf{DG = 4}$

Find BG:

$BG = \sqrt{BD^2 + DG^2}$ (Pythagorean Theorem)

Substitute

$BG = \sqrt{11^2 + 4^2} \\\\BG = \sqrt{137} \\\\\mathbf{BG = 11.7}$

Find BE:

$BE = \sqrt{BG^2 - EG^2}$ (Pythagorean Theorem)

Substitute

$BE = \sqrt{11.7^2 - 4^2} \\\\BE = \sqrt{120.89} \\\\\mathbf{BE = 10.99}$

Find GC:

$GC = \sqrt{CF^2 + FG^2}$ (Pythagorean Theorem)

Substitute

$GC = \sqrt{20^2 + 4^2} \\\\GC = \sqrt{416} \\\\\mathbf{GC = 20.4}$

In summary, applying the definition of the incenter of a triangle, the measures in triangle ABC where G is the incenter are:

$\mathbf{m \angle ABG = 20^{\circ}}$ $\mathbf{m \angle BCA = 22^{\circ}}$ $\mathbf{m \angle BAC = 118^{\circ}}$ $\mathbf{m \angle BAG = 59^{\circ}}$ DG = 4BE = 10.99BG = 11.7GC = 20.4

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Answer from: chelsiem25

m∠ABG = 20 degrees

m∠BCA = 22 degrees

m∠BAC = 118 degrees

m∠BAG = 59 degrees

DG = 4

BE = 12.4

BG = 11.7

GC = 20.4

Step-by-step explanation: