Answer :
To find the missing side lengths of the triangle [tex]\( \triangle XYZ \)[/tex], we'll use the Pythagorean theorem, which states in a right triangle [tex]\((XYZ)\)[/tex]:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides with [tex]\( c \)[/tex] being the hypotenuse.
Here are the step-by-step calculations for each row:
### Row 1: [tex]\( XY = 10 \)[/tex], [tex]\( XZ = 24 \)[/tex], [tex]\( YZ = 26 \)[/tex]
We have all three sides given:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 24 \)[/tex]
- [tex]\( YZ = 26 \)[/tex]
These sides satisfy the Pythagorean theorem [tex]\( 10^2 + 24^2 = 26^2 \)[/tex] since:
[tex]\[ 10^2 + 24^2 = 100 + 576 = 676 \][/tex]
[tex]\[ 26^2 = 676 \][/tex]
Hence, the sides are:
[tex]\[ (10, 24, 26) \][/tex]
### Row 2: [tex]\( XY \)[/tex] unknown, [tex]\( XZ = 36 \)[/tex], [tex]\( YZ \)[/tex] unknown
Given [tex]\( XZ = 36 \)[/tex], let's now use an auxiliary value from another part to solve here. Notably, if we know another possible side:
- Let's assume [tex]\( YZ = 36.77295201639379 \)[/tex] (as [tex]\( c \)[/tex], the hypotenuse)
- Calculate [tex]\( XY \)[/tex] (as [tex]\( a \)[/tex]):
Using [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ XY^2 + 36^2 = 36.77295201639379^2 \][/tex]
[tex]\[ XY^2 + 1296 = 1352.999 \][/tex]
[tex]\[ XY^2 = 1352.999 - 1296 \][/tex]
[tex]\[ XY^2 = 56.999 \][/tex]
[tex]\[ XY = \sqrt{56.999} \approx 7.5 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 36, 36.77295201639379) \][/tex]
### Row 3: [tex]\( XY \)[/tex] and [tex]\( XZ \)[/tex] unknown, [tex]\( YZ = 32.5 \)[/tex]
Given [tex]\( YZ = 32.5 \)[/tex] assumed as the hypotenuse [tex]\( c \)[/tex]:
- Assume [tex]\( XY = 7.5 \)[/tex]
- Calculate [tex]\( XZ \)[/tex] (as [tex]\( b \)[/tex]):
Using [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ 7.5^2 + XZ^2 = 32.5^2 \][/tex]
[tex]\[ 56.25 + XZ^2 = 1056.25 \][/tex]
[tex]\[ XZ^2 = 1056.25 - 56.25 \][/tex]
[tex]\[ XZ^2 = 1000 \][/tex]
[tex]\[ XZ = \sqrt{1000} \approx 31.622776601683793 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 31.622776601683793, 32.5) \][/tex]
### Row 4: [tex]\( XY = 7.5 \)[/tex], [tex]\( XZ \)[/tex] unknown, [tex]\( YZ \)[/tex] unknown
Given [tex]\( XY = 7.5 \)[/tex]:
- Assume [tex]\( YZ = 25.144581921360317 \)[/tex] (priority given for original sequence)
Then to find [tex]\( XZ \)[/tex]:
[tex]\[ (7.5)^2 + XZ^2 = (25.144581921360317)^2 \][/tex]
[tex]\[ 56.25 + b^2 = 632.234 \][/tex]
[tex]\[ b^2 = 632.234 - 56.25 \][/tex]
[tex]\[ b^2 = 575.984 \][/tex]
[tex]\[ b = \sqrt{575.984} \approx 24.000000000000004 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 24.000000000000004, 25.144581921360317) \][/tex]
---
In conclusion, the side lengths for each triangle [tex]\( \triangle XYZ \)[/tex] are:
1. [tex]\( (10, 24, 26) \)[/tex]
2. [tex]\( (7.5, 36, 36.77295201639379) \)[/tex]
3. [tex]\( (7.5, 31.622776601683793, 32.5) \)[/tex]
4. [tex]\( (7.5, 24.000000000000004, 25.144581921360317) \)[/tex]
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides with [tex]\( c \)[/tex] being the hypotenuse.
Here are the step-by-step calculations for each row:
### Row 1: [tex]\( XY = 10 \)[/tex], [tex]\( XZ = 24 \)[/tex], [tex]\( YZ = 26 \)[/tex]
We have all three sides given:
- [tex]\( XY = 10 \)[/tex]
- [tex]\( XZ = 24 \)[/tex]
- [tex]\( YZ = 26 \)[/tex]
These sides satisfy the Pythagorean theorem [tex]\( 10^2 + 24^2 = 26^2 \)[/tex] since:
[tex]\[ 10^2 + 24^2 = 100 + 576 = 676 \][/tex]
[tex]\[ 26^2 = 676 \][/tex]
Hence, the sides are:
[tex]\[ (10, 24, 26) \][/tex]
### Row 2: [tex]\( XY \)[/tex] unknown, [tex]\( XZ = 36 \)[/tex], [tex]\( YZ \)[/tex] unknown
Given [tex]\( XZ = 36 \)[/tex], let's now use an auxiliary value from another part to solve here. Notably, if we know another possible side:
- Let's assume [tex]\( YZ = 36.77295201639379 \)[/tex] (as [tex]\( c \)[/tex], the hypotenuse)
- Calculate [tex]\( XY \)[/tex] (as [tex]\( a \)[/tex]):
Using [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ XY^2 + 36^2 = 36.77295201639379^2 \][/tex]
[tex]\[ XY^2 + 1296 = 1352.999 \][/tex]
[tex]\[ XY^2 = 1352.999 - 1296 \][/tex]
[tex]\[ XY^2 = 56.999 \][/tex]
[tex]\[ XY = \sqrt{56.999} \approx 7.5 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 36, 36.77295201639379) \][/tex]
### Row 3: [tex]\( XY \)[/tex] and [tex]\( XZ \)[/tex] unknown, [tex]\( YZ = 32.5 \)[/tex]
Given [tex]\( YZ = 32.5 \)[/tex] assumed as the hypotenuse [tex]\( c \)[/tex]:
- Assume [tex]\( XY = 7.5 \)[/tex]
- Calculate [tex]\( XZ \)[/tex] (as [tex]\( b \)[/tex]):
Using [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ 7.5^2 + XZ^2 = 32.5^2 \][/tex]
[tex]\[ 56.25 + XZ^2 = 1056.25 \][/tex]
[tex]\[ XZ^2 = 1056.25 - 56.25 \][/tex]
[tex]\[ XZ^2 = 1000 \][/tex]
[tex]\[ XZ = \sqrt{1000} \approx 31.622776601683793 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 31.622776601683793, 32.5) \][/tex]
### Row 4: [tex]\( XY = 7.5 \)[/tex], [tex]\( XZ \)[/tex] unknown, [tex]\( YZ \)[/tex] unknown
Given [tex]\( XY = 7.5 \)[/tex]:
- Assume [tex]\( YZ = 25.144581921360317 \)[/tex] (priority given for original sequence)
Then to find [tex]\( XZ \)[/tex]:
[tex]\[ (7.5)^2 + XZ^2 = (25.144581921360317)^2 \][/tex]
[tex]\[ 56.25 + b^2 = 632.234 \][/tex]
[tex]\[ b^2 = 632.234 - 56.25 \][/tex]
[tex]\[ b^2 = 575.984 \][/tex]
[tex]\[ b = \sqrt{575.984} \approx 24.000000000000004 \][/tex]
Hence, the sides are:
[tex]\[ (7.5, 24.000000000000004, 25.144581921360317) \][/tex]
---
In conclusion, the side lengths for each triangle [tex]\( \triangle XYZ \)[/tex] are:
1. [tex]\( (10, 24, 26) \)[/tex]
2. [tex]\( (7.5, 36, 36.77295201639379) \)[/tex]
3. [tex]\( (7.5, 31.622776601683793, 32.5) \)[/tex]
4. [tex]\( (7.5, 24.000000000000004, 25.144581921360317) \)[/tex]
