Question

I need some help thanks

Answer 1

UV = 10

m∠UCT = 44

m∠UAV = 20

Explanation:

The angle bisectors of a triangle point of concurrency are called an incenter.

The properties of the incenter of a triangle are that the incenter is equidistant from the sides of the triangle (a segment connected from the incenter to the side of the triangle perpendicularly).

As previously mentioned, the angle bisectors form the incenter which means they bisect all three angles of a triangle.

The incenter of any triangle is always found on the inside.

So given that segment SV = 10 (this is the length of the side from the incenter to the side of a triangle perpendicularly as indicated by the right angles) this must mean all the other sides connected from the incenter to the side of a triangle must be equal to this too. So if we were trying to find the value of segment UV, we would apply the value of that segment to this one as this segment is also a side connected from the incenter to the side of the triangle.

Therefore UV = 10.

If we were given the angle of 22 as an angle bisector of angle UCT, all we would need to do to find the measure of angle UCT is double it.

So 22 x 2 = 44, thus measure of angle UCT is equal to 44 degrees.

Given two angle measures out of the entire triangle, triangle ABC, we know that one angle is equal to 96 degrees (given by measure of angle SBT is equal to 96), another 44, and the third is unknown (the angle measure we are trying to find out).

According to the triangle angle sum theorem, the sum of the interior angles of a triangle is always 180 degrees.

So we just add these two given angles plus the unknown angle (angle BAC) and set it to equal 180.

Let's execute this:

96 + 44 + x = 180

Solve for x:

140 + x = 180

-140 -140

x = 40, so the angle measure of BAC is 40.

However, they want us to find the angle that bisects BAC, which is angle UAV.

Divide 40 by 2 to find one angle measure of the bisected angle (BAC):

40/2

= 20, the measure of angle UAV is 20 degrees.

Answer 2

UV = 16.9

mUCT = 269.2°

mUAV = 35.8°

Finding UV:

We can use the Angle Bisector Theorem to find UV.

This theorem states that the ratio of the sides of a triangle is equal to the ratio of the segments created by an angle bisector on that side.

In this case, we have:

BV = 13 (given)

VS = 10 (given)

UV = x (unknown)

The ratio of BV to VS is equal to the ratio of the whole side AB to AV. We know that AV = BV + VS = 13 + 10 = 23.

Therefore:

BV/VS = AB/AV

13/10 = AB/23

AB = (13/10) * 23

AB = 29.9

Now we can find UV:

UV = AB - BV

UV = 29.9 - 13

UV = 16.9

Therefore, UV = 16.9.

Finding mUCT:

We can use the Angle Bisector Theorem again to find mUCT. This time, we will use the sides CV and UV:

CV = (2 * BV * VS) / (BV + VS)

CV = (2 * 13 * 10) / (13 + 10)

CV = 260 / 23

CV = 11.3

Now we can find mUCT:

mUCT = (UV / CV) * 180°

mUCT = (16.9 / 11.3) * 180°

mUCT = 269.2°

Therefore, mUCT = 269.2°.

Finding mUAV:

We can use the Angle Bisector Theorem one last time to find mUAV. This time, we will use the sides AV and UV:

AV = (2 * BV * VS) / (BV - VS)

AV = (2 * 13 * 10) / (13 - 10)

AV = 260 / 3

AV = 86.7

Now we can find mUAV:

mUAV = (UV / AV) * 180°

mUAV = (16.9 / 86.7) * 180°

mUAV = 35.8°

Therefore, mUAV = 35.8°.