1. given that ps = 53.4, qt = 47.7, and qs = 53.4, find pq

2 given that m is perpendicular bisector of $\overline{pq}$ and sq = 25.9, find sp

3. given that m is perpendicular bisector of $\overline{pq}$, ps = 4a, and qs = 2a + 26, find qs.

use the diagram to answer the questions below.
4. given that $\overline{bd}$ bisects $\angle abc$ and cd = 21.9, find ad.
5. given that ad = 61, cd = 61, and $m\angle abc = 48^\circ$, find $m\angle cbd$.
6. given that da = cd, $m\angle dbc = (10x + 3)^\circ$, and $m\angle dba = (8x + 10)^\circ$, find $m\angle dbc$.

Question

1. given that ps = 53.4, qt = 47.7, and qs = 53.4, find pq

2 given that m is perpendicular bisector of $\overline{pq}$ and sq = 25.9, find sp

3. given that m is perpendicular bisector of $\overline{pq}$, ps = 4a, and qs = 2a + 26, find qs.

use the diagram to answer the questions below.
4. given that $\overline{bd}$ bisects $\angle abc$ and cd = 21.9, find ad.
5. given that ad = 61, cd = 61, and $m\angle abc = 48^\circ$, find $m\angle cbd$.
6. given that da = cd, $m\angle dbc = (10x + 3)^\circ$, and $m\angle dba = (8x + 10)^\circ$, find $m\angle dbc$.

Accepted Answer

### Problem 1
# Explanation:
## Step1: Analyze triangle properties
Since $ PS = QS = 53.4 $, triangle $ PSQ $ is isosceles with $ PT = QT $ (as $ m $ is perpendicular to $ PQ $).
## Step2: Calculate $ PQ $
$ PQ = 2\times QT = 2\times47.7 = 95.4 $
# Answer:
$ 95.4 $

### Problem 2
# Explanation:
## Step1: Use perpendicular bisector property
A perpendicular bisector of a segment implies that any point on it is equidistant from the endpoints of the segment. So $ SP = SQ $.
## Step2: Substitute $ SQ $ value
Given $ SQ = 25.9 $, so $ SP = 25.9 $
# Answer:
$ 25.9 $

### Problem 3
# Explanation:
## Step1: Apply perpendicular bisector property
Since $ m $ is the perpendicular bisector of $ PQ $, $ PS = QS $. So $ 4a=2a + 26 $
## Step2: Solve for $ a $
$ 4a-2a=26 $
$ 2a = 26 $
$ a = 13 $
## Step3: Calculate $ QS $
Substitute $ a = 13 $ into $ QS = 2a + 26 $
$ QS=2\times13 + 26=26 + 26 = 52 $
# Answer:
$ 52 $

### Problem 4
# Explanation:
## Step1: Use angle - bisector property
The angle - bisector theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. Also, if a point is on the angle bisector, it is equidistant from the sides of the angle. Since $ BD $ bisects $ \angle ABC $ and $ AD\perp BA $, $ CD\perp BC $, we have $ AD = CD $
## Step2: Substitute $ CD $ value
Given $ CD = 21.9 $, so $ AD = 21.9 $
# Answer:
$ 21.9 $

### Problem 5
# Explanation:
## Step1: Analyze distance and angle - bisector
Since $ AD = CD = 61 $ and $ AD\perp BA $, $ CD\perp BC $, by the converse of the angle - bisector theorem, $ BD $ bisects $ \angle ABC $
## Step2: Calculate $ m\angle CBD $
Given $ m\angle ABC=48^{\circ} $, then $ m\angle CBD=\frac{1}{2}m\angle ABC=\frac{1}{2}\times48^{\circ} = 24^{\circ} $
# Answer:
$ 24^{\circ} $

### Problem 6
# Explanation:
## Step1: Use angle - bisector property
Since $ DA = CD $ and $ AD\perp BA $, $ CD\perp BC $, $ BD $ bisects $ \angle ABC $, so $ m\angle DBC=m\angle DBA $
## Step2: Set up equation
$ 10x + 3=8x + 10 $
## Step3: Solve for $ x $
$ 10x-8x=10 - 3 $
$ 2x = 7 $
$ x=\frac{7}{2}=3.5 $
## Step4: Calculate $ m\angle DBC $
Substitute $ x = 3.5 $ into $ m\angle DBC=(10x + 3)^{\circ} $
$ m\angle DBC=10\times3.5+3=35 + 3=38^{\circ} $
# Answer:
$ 38^{\circ} $

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QUESTION IMAGE

Question

Explanation:

Problem 1

Step1: Analyze triangle properties

Step2: Calculate \( PQ \)

Step1: Use perpendicular bisector property

Step2: Substitute \( SQ \) value

Step1: Apply perpendicular bisector property

Step2: Solve for \( a \)

Step3: Calculate \( QS \)

Answer:

Problem 2